Voyager I

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import math as m
from scipy.optimize import fsolve, brentq, newton, least_squares, fmin
from matplotlib.lines import Line2D
from matplotlib import colors
from scipy.interpolate import griddata
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.ticker import ScalarFormatter
import time
import warnings
from tqdm import tqdm
import pickle

from astropy import units as u
from astropy.time import Time
from astropy.coordinates import solar_system_ephemeris
solar_system_ephemeris.set("jpl")

from poliastro.bodies import *
from poliastro.ephem import Ephem
from poliastro.frames import Planes
from poliastro.maneuver import Maneuver
from poliastro.twobody import Orbit
from poliastro.util import norm, time_range
from poliastro.plotting.static import StaticOrbitPlotter

from itertools import permutations, combinations
from random import shuffle
import random
import statistics
import seaborn as sns
from IPython.display import HTML
import matplotlib.animation as animation

from poliastro.core.iod import vallado
from poliastro.core.elements import rv2coe
import datetime

import os
from pathlib import Path

from IPython.display import HTML, display
from matplotlib.animation import PillowWriter
import io
from base64 import b64encode

def show(variable, units):
    def get_var_name(var):
        for name, value in globals().items():
            if value is var:
                return name
    print(get_var_name(variable), "=", variable, units)

plt.rc('text', usetex=True)
plt.rc('font', family='serif')

# Create an output directory handler
class OutputManager:
    def __init__(self, base_dir="V1"):
        # Initialise the output manager with a base directory
        self.base_dir = Path(base_dir)
        self.base_dir.mkdir(exist_ok=True)
        print(f"Output directory set to: {self.base_dir.absolute()}")

    # Save a figure with consistent settings
    def save_file(self, filename, dpi=500, bbox_inches="tight", **kwargs):
        # Ensure the filename has an extension
        if not any(filename.endswith(ext) for ext in ['.png', '.jpg', '.pdf', '.svg']):
            filename = f"{filename}.png"
        
        # Create the full path
        full_path = self.base_dir / filename
        
        # Save the figure
        plt.savefig(full_path, dpi=dpi, bbox_inches=bbox_inches, **kwargs)
        print(f"Figure saved to: {full_path}")
        
        return full_path

    # Get the full path for a file.
    def get_path(self, filename):
        return self.base_dir / filename
 
# Create an instance of the output manager
output_figures = OutputManager("Voyager 1 - Figures")
output_pickle = OutputManager("Raw Results")
output_ani = OutputManager("Voyager 1 - Animations")

Output directory set to: /Users/snath2/Desktop/Aero & Astro/Year 3/IP/Final IP/Optimisation Algorithms/Voyager 1 - Figures
Output directory set to: /Users/snath2/Desktop/Aero & Astro/Year 3/IP/Final IP/Optimisation Algorithms/Raw Results
Output directory set to: /Users/snath2/Desktop/Aero & Astro/Year 3/IP/Final IP/Optimisation Algorithms/Voyager 1 - Animations

global_flag = False

mu_earth = Earth.k.to(u.km**3/u.s**2).value
mu_jupiter = Jupiter.k.to(u.km**3/u.s**2).value
mu_saturn = Saturn.k.to(u.km**3/u.s**2).value
mu_sun = Sun.k.to(u.km**3/u.s**2).value
mu_sun

132712442099.00002

mu = mu_sun # Sun's gravitational parameter (km³/s²)
mu

132712442099.00002

# Define parula colormap (MATLAB-style) with an option to reverse
def parula_map(reverse=False):
    # RGB values
    cm_data = [
        [0.2081, 0.1663, 0.5292], [0.2116, 0.1898, 0.5777], [0.2123, 0.2138, 0.6270],
        [0.2081, 0.2386, 0.6771], [0.1959, 0.2645, 0.7279], [0.1707, 0.2919, 0.7792],
        [0.1253, 0.3242, 0.8303], [0.0591, 0.3598, 0.8683], [0.0117, 0.3875, 0.8820],
        [0.0060, 0.4086, 0.8828], [0.0165, 0.4266, 0.8786], [0.0329, 0.4430, 0.8720],
        [0.0498, 0.4586, 0.8641], [0.0629, 0.4737, 0.8554], [0.0723, 0.4887, 0.8467],
        [0.0779, 0.5040, 0.8384], [0.0793, 0.5200, 0.8312], [0.0749, 0.5375, 0.8263],
        [0.0641, 0.5570, 0.8240], [0.0488, 0.5772, 0.8228], [0.0343, 0.5966, 0.8199],
        [0.0265, 0.6137, 0.8135], [0.0239, 0.6287, 0.8038], [0.0231, 0.6418, 0.7913],
        [0.0228, 0.6535, 0.7768], [0.0267, 0.6642, 0.7607], [0.0384, 0.6743, 0.7436],
        [0.0590, 0.6838, 0.7254], [0.0843, 0.6928, 0.7062], [0.1133, 0.7015, 0.6859],
        [0.1453, 0.7098, 0.6646], [0.1801, 0.7177, 0.6424], [0.2178, 0.7250, 0.6193],
        [0.2586, 0.7317, 0.5954], [0.3022, 0.7376, 0.5712], [0.3482, 0.7424, 0.5473],
        [0.3953, 0.7459, 0.5244], [0.4420, 0.7481, 0.5033], [0.4871, 0.7491, 0.4840],
        [0.5300, 0.7491, 0.4661], [0.5709, 0.7485, 0.4494], [0.6099, 0.7473, 0.4337],
        [0.6473, 0.7456, 0.4188], [0.6834, 0.7435, 0.4044], [0.7184, 0.7411, 0.3905],
        [0.7525, 0.7384, 0.3768], [0.7858, 0.7356, 0.3633], [0.8185, 0.7327, 0.3498],
        [0.8507, 0.7299, 0.3360], [0.8824, 0.7274, 0.3217], [0.9139, 0.7258, 0.3063],
        [0.9450, 0.7261, 0.2886], [0.9739, 0.7314, 0.2666], [0.9938, 0.7455, 0.2403],
        [0.9990, 0.7653, 0.2164], [0.9955, 0.7861, 0.1967], [0.9880, 0.8066, 0.1794],
        [0.9789, 0.8271, 0.1633], [0.9697, 0.8481, 0.1475], [0.9626, 0.8705, 0.1309],
        [0.9589, 0.8949, 0.1132], [0.9598, 0.9218, 0.0948], [0.9661, 0.9514, 0.0755],
        [0.9763, 0.9831, 0.0538]
    ]
    
    if reverse:
        cm_data = cm_data[::-1]  # Reverse the list if specified
        
    return colors.ListedColormap(cm_data)

1) Choose the epochs $T_1$ to $T_N$, corresponding to all the planets $P_1$ to $P_N$ in the mission. This is decided later in the optimisation algorithm.

#Voyager I real mission timing
scale = "tt"

date_departure = date_earth_jupiter = "1977-09-08 09:08"

date_flyby_jupiter = "1979-03-05 12:05"
#date_jupiter_saturn = "1979-04-24 07:33:03"

date_arrival = date_flyby_saturn = "1980-11-12 23:46"

1.1) Convert epochs to MJD

from astropy.time import Time
# e.g T_et = "1979-03-05 12:05:26"
def convert_ET(T_et):
    # Defines time scale for astropy.time of ET, which approx equal to TT 
    T_tt = Time(T_et, scale='tt')
    # Calculate TDB (only 1-2 milliseconds different from ET/TT)
    T_tdb = T_tt.tdb
    # Calculates the TAI (T_tai = T_tt - 32.184s)
    T_tai = T_tt.tai
    # Calculates the JD from epoch 1st January 4713 BC at 12:00 TT
    T_jd = T_tt.jd 
    # Calculates MJD
    T_mjd = T_jd - 2400000.5 
    # Calculate UTC using NASA's Sprice Time Subsystem
    T_utc = T_tt.utc
    return {"TDB": T_tdb, "TAI": T_tai, "JD": T_jd, "MJD": T_mjd, "UTC": T_utc}

scale = "mjd"

date_departure_mjd = convert_ET(date_departure)['MJD']

date_flyby_jupiter_mjd = convert_ET(date_flyby_jupiter)['MJD']

date_flyby_saturn_mjd = convert_ET(date_flyby_saturn)['MJD']

show(date_departure_mjd,"MJD")
show(date_flyby_jupiter_mjd,"MJD")
show(date_flyby_saturn_mjd,"MJD")

date_departure_mjd = 43394.38055555569 MJD
date_flyby_jupiter_mjd = 43937.503472222015 MJD
date_flyby_saturn_mjd = 44555.99027777789 MJD

1.2) Find the $\Delta t$ for each mission trajectory

def find_tof(start_date_days, end_date_days):
    delta_t = (end_date_days - start_date_days) * (60*60*24)
    return delta_t

delta_t_earth_jupiter = find_tof(date_departure_mjd, date_flyby_jupiter_mjd)
delta_t_jupiter_saturn = find_tof(date_flyby_jupiter_mjd, date_flyby_saturn_mjd)
print("Earth-Jupiter:",delta_t_earth_jupiter,"s")
print("Jupiter-Saturn:",delta_t_jupiter_saturn,"s")

Earth-Jupiter: 46925819.999970496 s
Jupiter-Saturn: 53437260.000027716 s

2) Using the known epochs $T_1$ to $T_N$, calculate their respective planetary state vectors $\{\vec{\mathbf{r}},\vec{\mathbf{V}}^{\,(\text{P})}\}$ using Algorithm 3.3.

def get_planet_state_vector(planet, date_time, scale):
    # Convert the date_time in MJD to modern TDB time scale 
    date = Time(date_time, format='mjd', scale=scale).tdb
    # Generate the planet's ephemeris at the given time
    planet_ephem = Ephem.from_body(planet, date, plane=Planes.EARTH_ECLIPTIC)
    # Extract the position and velocity vectors
    r, v = planet_ephem.rv(date)
    r = r.to(u.km)
    v = v.to(u.km/u.s) 
    return r, v

scale = 'tt'
r_earth, v_earth = get_planet_state_vector(Earth, date_departure_mjd, scale)
r_jupiter, v_jupiter = get_planet_state_vector(Jupiter, date_flyby_jupiter_mjd, scale)
r_saturn, v_saturn = get_planet_state_vector(Saturn, date_flyby_saturn_mjd, scale)

print("Earth:")
display(r_earth, v_earth)
print("Jupiter:")
display(r_jupiter, v_jupiter)
print("Saturn:")
display(r_saturn, v_saturn)

Earth:

$[1.4652729 \times 10^{8},~-37326667,~-10360.24] \; \mathrm{km}$

$[6.7884394,~28.786762,~0.0010233513] \; \mathrm{\frac{km}{s}}$

Jupiter:

$[-4.8121511 \times 10^{8},~6.2707394 \times 10^{8},~8194396.5] \; \mathrm{km}$

$[-10.51368,~-7.3532518,~0.26572085] \; \mathrm{\frac{km}{s}}$

Saturn:

$[-1.4200499 \times 10^{9},~-54308565,~57383062] \; \mathrm{km}$

$[-0.14811167,~-9.6697442,~0.17492744] \; \mathrm{\frac{km}{s}}$

r_earth, v_earth = r_earth.value, v_earth.value
r_jupiter, v_jupiter = r_jupiter.value, v_jupiter.value
r_saturn, v_saturn = r_saturn.value, v_saturn.value

2.1) Find the Real Mission Velocities

import numpy as np
from scipy.optimize import fsolve

# Known parameters K = [e,a,i,Omega,w,M]
mu = 1.327e11  # Sun's gravitational parameter (km^3/s^2)


# Compute true anomaly (theta) for 0<e<1 (elliptical orbit)
def elliptic_theta(M, e):
    def find_E(E):
        return E - e * np.sin(E) - M

    E = fsolve(find_E, 0.01)[0] # Eccentric Anomaly
    theta = 2 * np.arctan(np.tan(E / 2) * np.sqrt((1 + e) / (1 - e)))
    return theta


# Compute true anomaly (theta)  for e=1 (parabolic orbit)
def parabolic_theta(M):
    var = 3 * M + np.sqrt(9 * M**2 + 1)
    B = var ** (1 / 3) - var ** (-1 / 3) # Barker Parameter
    theta = 2 * np.arctan(B)
    return theta


# Compute true anomaly (theta)  for e>1 (hyperbolic orbit)
def hyperbolic_theta(M, e):
    def find_F(F):
        return e * np.sinh(F) - F - M

    F = fsolve(find_F, 0.01)[0] # Hyperbolic Eccentric Anomaly
    theta = 2 * np.arctan(np.tanh(F / 2) * np.sqrt((e + 1) / (e - 1)))
    return theta


def forwardTransformation(e, a, i, Omega, w, M, mu, p=None):
    # Compute specific angular momentum h and radius r from theta
    p = (a * (1 - e**2) if p is None else p)
    h = np.sqrt(mu * p)

    if e < 1:
        theta = elliptic_theta(M, e)
    elif e == 1:
        theta = parabolic_theta(M)
    else:
        theta = hyperbolic_theta(M, e)

    r = p / (1 + e * np.cos(theta))

    # Compute position and velocity in perifocal coordinates
    r_perifocal = np.array([r * np.cos(theta), r * np.sin(theta), 0])
    v_perifocal = (mu / h) * np.array([-np.sin(theta), e + np.cos(theta), 0])

    # Compute transformation matrix Q
    cos_Omega, sin_Omega = np.cos(Omega), np.sin(Omega)
    cos_w, sin_w = np.cos(w), np.sin(w)
    cos_i, sin_i = np.cos(i), np.sin(i)
    Q = np.array(
        [
            [
                cos_Omega * cos_w - sin_Omega * sin_w * cos_i,
                -cos_Omega * sin_w - sin_Omega * cos_i * cos_w,
                sin_Omega * sin_i,
            ],
            [
                sin_Omega * cos_w + cos_Omega * cos_i * sin_w,
                -sin_Omega * sin_w + cos_Omega * cos_i * cos_w,
                -cos_Omega * sin_i,
            ],
            [sin_i * sin_w, sin_i * cos_w, cos_i],
        ]
    )

    # Transform to reference frame
    r_reference = np.dot(Q, r_perifocal)
    v_reference = np.dot(Q, v_perifocal)
    return r_reference, v_reference

3) Now, to define the transfer trajectory we separate them into legs: $P_1\text{-}P_2$, $P_2\text{-}P_3$, ..., $P_{N-1}\text{-}P_N$. For each leg, we can use the solution to Lambert's Problem (Algorithm 3.4) to get its transfer velocity vectors. This fully defines each leg, where for instance $P_1\text{-}P_2$ leg, we have the following parameters $\big\{\vec{\mathbf{r}}_{1}, \vec{\mathbf{r}}_{2}, \vec{\mathbf{V}}^{\,(\text{P})}_1,\vec{\mathbf{V}}^{\,(\text{P})}_2, (\vec{\mathbf{v}})_{D}, (\vec{\mathbf{v}}_{T_2})_{A}, \Delta t\big\}$.

# from poliastro.core.iod import vallado

# # Lambert Solver
# def lambert(r1, r2, delta_t):
#     v_d, v_a = vallado(mu_sun, r1, r2, delta_t, M=0, prograde=True, lowpath=True, numiter=1e6, rtol=1e-8)
#     return v_d, v_a

import numpy as np
from scipy.optimize import brentq, newton, least_squares

# Known parameters [r1, r2, delta_t]
mu = 1.327e11  # Sun's gravitational parameter (km^3/s^2)


# Stumpff functions
def C(z):
    if z > 0:
        return (1 - np.cos(np.sqrt(z))) / z
    elif z < 0:
        return (np.cosh(np.sqrt(-z)) - 1) / (-z)
    else:
        return 0.5


def S(z):
    if z > 0:
        return (np.sqrt(z) - np.sin(np.sqrt(z))) / (z**1.5)
    elif z < 0:
        return (np.sinh(np.sqrt(-z)) - np.sqrt(-z)) / ((-z) ** 1.5)
    else:
        return 1 / 6


# Compute y parameter
def compute_y(z, r1, r2, A):
    C_z = max(C(z), 1e-8)  # Ensure no division by zero
    return max(r1 + r2 + A * (z * S(z) - 1) / np.sqrt(C_z), 1e-8)


# Time-of-flight equation
def F_lambert(z, delta_t, r1, r2, A):
    y_val = compute_y(z, r1, r2, A)
    return ((y_val / C(z)) ** 1.5) * S(z) + A * np.sqrt(y_val) - np.sqrt(mu) * delta_t


# Lambert solver
def lambert(r1, r2, deltat):
    # Step 1: Find the delta_theta between P1 and P2
    r1_mag, r2_mag = np.linalg.norm(r1), np.linalg.norm(r2)
    delta_theta = np.arccos(np.dot(r1, r2) / (r1_mag * r2_mag))

    if np.cross(r1, r2)[2] < 0:
        delta_theta = 2 * np.pi - delta_theta

    # Step 2: Solve for the intermediate auxiliary variable z
    A = np.sin(delta_theta) * np.sqrt(r1_mag * r2_mag / (1 - np.cos(delta_theta)))

    def func(z_val):
        return F_lambert(z_val, deltat, r1_mag, r2_mag, A)

    def solver(func, z_min=-1e3, z_max=1e3, z_guess=0.01):
        try:
            z = brentq(lambda z: func(z), z_min, z_max)
        except ValueError:
            try:
                z = newton(func, z_guess)
            except RuntimeError:
                z = least_squares(func, z_guess).x[0]
        return z

    z = solver(func)
    y_val = compute_y(z, r1_mag, r2_mag, A)

    # Step 3: Compute Lagrange coefficients f, g, gdot
    f = 1 - y_val / r1_mag
    g = A * np.sqrt(y_val / mu)
    gdot = 1 - y_val / r2_mag

    v1 = (r2 - f * r1) / g
    v2 = (gdot * r2 - r1) / g
    return v1, v2

v1_d, v2_a = lambert(r_earth, r_jupiter, delta_t_earth_jupiter)
v2_d, v3_a = lambert(r_jupiter, r_saturn, delta_t_jupiter_saturn)
print(v1_d, v2_a)
print(v2_d, v3_a)

[10.72074241 38.28749663  0.66273895] [-1.22415214e+01  3.46208225e+00  6.42429222e-03]
[-20.97912678 -10.58575624   1.01828126] [-14.94072746 -13.42269638   0.82774958]

print(np.linalg.norm(v1_d), np.linalg.norm(v2_a))
print(np.linalg.norm(v2_d), np.linalg.norm(v3_a))

39.765637662752795 12.721670500594671
23.520605691058012 20.101723417599942

4) We can now work out $\Delta V_{injection}$, the initial injection burn from $P_1$ into transfer orbit,

deltaV_injection = v1_d - v_earth
deltaV_injection = np.linalg.norm(deltaV_injection)
deltaV_injection

10.303632164583048

# Isp = 287.3 #s
# g0 = 9.80665 #m/s^2

# Vex = Isp*g0 / 1000 # km/s
# Vex, 'km/s'

# M_0_propulsion_module = 1123 #kg
# M_f_capacity_propulsion_module = 1039 #kg

# M_after_injection = M_0_propulsion_module*( 1 - np.exp(-deltaV_injection/Vex) )
# M_after_injection

# deltaV = deltaV_injection
# M0 = M_0_propulsion_module
# Mf = M_f_capacity_propulsion_module

# def rocket_eq(Vex):
#     return Vex*np.log(M0/(M0 - Mf)) - deltaV

# Vex = fsolve(rocket_eq, 0.01)[0]
# Vex, 'km/s'

Flyby

5) Find $\Delta V_{p\,{(flyby)}}$, we must first find the hyperbolic excess velocities $\{\vec{\mathbf{v}}_{\infty_{in}},\vec{\mathbf{v}}_{\infty_{out}} \}$ at each flyby planet

6) Find the planet's inbound $a_{in}$ and outbound $a_{out}$ semi-major axis

7) Find its deflection (turning) angle $\delta$ from entering SOI to exiting SOI

Here, we can utilise the equation relating deflection (turning) angle $\delta$ to the outbound eccentricity $e_{out}$ of hyperbolic trajectory,

8) Hence, the radius of the closest approach to the planet $r_{p\,{(flyby)}}$ can be calculated

9) We now have all the required parameter to equate $\Delta V_{p\,{(flyby)}}$,

R_jupiter = 69911 #km
R_saturn = 58232 #km

R_jupiter_SOI = 48.2e6 #km
R_saturn_SOI = 54.5e6 #km

# Add must be more than R_planet but less that r_SOI <-- NEED TO ADD!!!!!

def find_deltaV_flyby(v_in, v_out, v_planet, mu_planet, R_planet, R_SOI):
    vinf_in = v_in - v_planet
    vinf_out = v_out - v_planet

    a_in = - mu_planet/np.linalg.norm(vinf_in)**2
    a_out = - mu_planet/np.linalg.norm(vinf_out)**2

    deflection = np.arccos( np.dot(vinf_in,vinf_out)/( np.linalg.norm(vinf_in) * np.linalg.norm(vinf_out) ) )

    e_out = 1/np.sin(deflection/2)
    # def f(e_out):
    #     return (a_out/a_in * (e_out - 1))*np.sin( deflection - np.arcsin(1/e_out) ) - 1   
    # e_out = fsolve(f, 1.1)[0]

    rp = a_out*(1-e_out)

    if rp < R_planet and rp > R_SOI:  
        deltaV_flyby = 1e99
    else:
        vp_esc = np.sqrt(2*mu_planet/rp)
        
        vp_flyby_in = np.sqrt( np.linalg.norm(vinf_in)**2 + (2 * mu_planet)/rp )

        if vp_flyby_in > vp_esc:
            deltaV_flyby = abs(np.sqrt( np.linalg.norm(vinf_out)**2 + (2 * mu_planet)/rp ) - np.sqrt( np.linalg.norm(vinf_in)**2 + (2 * mu_planet)/rp ) )
        else:
            deltaV_flyby = 1e99

    return deltaV_flyby, rp

deltaV_jupiter, rp_jupiter = find_deltaV_flyby(v2_a,v2_d,v_jupiter,mu_jupiter,R_jupiter, R_jupiter_SOI)
deltaV_jupiter

0.008783673764401811

10) Finally, to summate all the individual $\Delta V$ components of the initial injection and subsequent flybys we can equate,

deltaV_total = deltaV_injection + deltaV_jupiter
deltaV_total

10.31241583834745

(date_departure_mjd, date_flyby_jupiter_mjd, date_flyby_saturn_mjd)

(43394.38055555569, 43937.503472222015, 44555.99027777789)

11) From the calculated mission $\Delta V$, we can find the fuel mass $M_f$ used.

NASA Data

M0_Voyager = 2066 #kg 

M0_propulsion_module = 1207 #kg
Mf_propulsion_module = 1039 #kg

M0_mission_module = 825 #kg
Mf_mission_module = 100 #kg

Mf_total_NASA = Mf_propulsion_module + Mf_mission_module

F = 68054 #N
t = 43 #s
g0 = 9.80665 # m/s2

# The Isp of the actual Voyager I mission - propulsion module
Isp_NASA = (F*t)/(Mf_propulsion_module*g0)
Isp_NASA, 's'

(287.2009612891239, 's')

Specific impulse ($I_{sp}$) is a measure of a rocket engine's efficiency, defined as the impulse (thrust per unit weight flow of propellant) delivered per unit of propellant consumed.

My Real Simulation Data

def find_Isp(Mf, M0, deltaV):
    Isp = (deltaV*1e3)/( np.log(M0/(M0 - Mf)) * g0)
    return Isp

Isp_my_sim = find_Isp(Mf_total_NASA, M0_Voyager, deltaV_total)
Isp_my_sim, 's'

(1312.1445693172518, 's')

The difference in $I_{sp}$ of NASA data and my_sim data is due to the inherit assumption in my model (Method of Patched Conics and Lambert Transfers) -- also the unpowered gravitational assist assumption.

Also, due to the fact NASA used a 2 stage system with the propulsion system ejected after injection into Earth-Jupiter transfer orbit. This left a 825kg mission module with around 100kg of fuel on it.

Vex = Isp_my_sim*g0 / 1000 # km/s
Vex, 'km/s'

(12.867742540695026, 'km/s')

M0_Voyager = M0 = 2066 #kg 

def compute_fuel_mass(M0, Vex, deltaV):
    Mf = M0*( 1 - np.exp(-deltaV/Vex) )
    return Mf 
    
fuel_mass_total = compute_fuel_mass(M0, Vex, deltaV_total)

print(f"The mission deltaV of {deltaV_total:.4f}km/s caused {fuel_mass_total:.2f}kg of fuel used.")

The mission deltaV of 10.3124km/s caused 1139.00kg of fuel used.

---

Function: find_deltaV_mission()

def find_deltaV_mission(date_departure_mjd, date_flyby_jupiter_mjd, date_flyby_saturn_mjd):
    #Step 1 - Find delta t
    delta_t_earth_jupiter = find_tof(date_departure_mjd, date_flyby_jupiter_mjd) #s
    delta_t_jupiter_saturn = find_tof(date_flyby_jupiter_mjd, date_flyby_saturn_mjd) #s
    tof = delta_t_earth_jupiter + delta_t_jupiter_saturn #s

    #Step 2 - Find planetary state vectors
    scale = 'tt'
    r_earth, v_earth = get_planet_state_vector(Earth, date_departure_mjd, scale)
    r_jupiter, v_jupiter = get_planet_state_vector(Jupiter, date_flyby_jupiter_mjd, scale)
    r_saturn, v_saturn = get_planet_state_vector(Saturn, date_flyby_saturn_mjd, scale)

    r_earth, v_earth = r_earth.value, v_earth.value
    r_jupiter, v_jupiter = r_jupiter.value, v_jupiter.value
    r_saturn, v_saturn = r_saturn.value, v_saturn.value

    #Step 3 - Define trajectory fully using Lambert Transfer
    v1_d, v2_a = lambert(r_earth, r_jupiter, delta_t_earth_jupiter)
    v2_d, v3_a = lambert(r_jupiter, r_saturn, delta_t_jupiter_saturn)
        
    #Step 4 - Injection deltaV
    deltaV_injection = v1_d - v_earth
    deltaV_injection = np.linalg.norm(deltaV_injection)
    
    #Step 5 - Find deltaV of flyby
    deltaV_jupiter, rp_jupiter = find_deltaV_flyby(v2_a,v2_d,v_jupiter,mu_jupiter,R_jupiter,R_jupiter_SOI)
    #if deltaV_jupiter == 1e99:
    #    tof = 1e99
    
    #Step 6 - Find total deltaV of mission
    deltaV_total = deltaV_injection + deltaV_jupiter

    #Step 7 - Find the fuel mass used
    fuel_mass_total = compute_fuel_mass(M0, Vex, deltaV_total)

    data_dict = {
        "deltaV_injection (km/s)": deltaV_injection,
        "deltaV_jupiter (km/s)": deltaV_jupiter,
        "rp_jupiter (km)": rp_jupiter,
        "tof (days)": tof/(60*60*24)
    }
       
    return deltaV_total, fuel_mass_total, data_dict

deltaV_REAL, fuel_mass_REAL, data_dict_REAL = find_deltaV_mission(date_departure_mjd, date_flyby_jupiter_mjd, date_flyby_saturn_mjd)
deltaV_REAL, fuel_mass_REAL, data_dict_REAL 

(10.31241583834745,
 1139.0,
 {'deltaV_injection (km/s)': 10.303632164583048,
  'deltaV_jupiter (km/s)': 0.008783673764401811,
  'rp_jupiter (km)': 339964.07869084046,
  'tof (days)': 1161.6097222222015})

dates_REAL = [date_departure_mjd, date_flyby_jupiter_mjd, date_flyby_saturn_mjd]
dates_REAL

[43394.38055555569, 43937.503472222015, 44555.99027777789]

---

Optimisation Algorithms

def MJD_to_TT_calander_date(T_mjd):
    T_TT = Time(T_mjd, format='mjd', scale='tt') #Time object: scale='tt' format='mjd'
    return T_TT.iso

MJD_to_TT_calander_date(date_departure_mjd)

'1977-09-08 09:08:00.000'

convert_ET( MJD_to_TT_calander_date(date_departure_mjd) )['UTC']

<Time object: scale='utc' format='iso' value=1977-09-08 09:07:11.816>

change_in_time_days = 365

Brute force Algorithm

1.1) Find the $\Delta t$ for each mission trajectory

print("Earth-Jupiter:",delta_t_earth_jupiter/(60*60*24),"days")
print("Jupiter-Saturn:",delta_t_jupiter_saturn/(60*60*24),"days")

Earth-Jupiter: 543.1229166663252 days
Jupiter-Saturn: 618.4868055558763 days

1.2) Define the range for each date allowed for algorithm

population = spacing = 50
print("No. of Iterations:",spacing**3)

No. of Iterations: 125000

time_range_LIST = np.linspace(-change_in_time_days, change_in_time_days, spacing)
time_range_LIST

array([-365.        , -350.10204082, -335.20408163, -320.30612245,
       -305.40816327, -290.51020408, -275.6122449 , -260.71428571,
       -245.81632653, -230.91836735, -216.02040816, -201.12244898,
       -186.2244898 , -171.32653061, -156.42857143, -141.53061224,
       -126.63265306, -111.73469388,  -96.83673469,  -81.93877551,
        -67.04081633,  -52.14285714,  -37.24489796,  -22.34693878,
         -7.44897959,    7.44897959,   22.34693878,   37.24489796,
         52.14285714,   67.04081633,   81.93877551,   96.83673469,
        111.73469388,  126.63265306,  141.53061224,  156.42857143,
        171.32653061,  186.2244898 ,  201.12244898,  216.02040816,
        230.91836735,  245.81632653,  260.71428571,  275.6122449 ,
        290.51020408,  305.40816327,  320.30612245,  335.20408163,
        350.10204082,  365.        ])

departure_time_range_LIST = np.linspace(-change_in_time_days, change_in_time_days, spacing)
departure_time_range_LIST

array([-365.        , -350.10204082, -335.20408163, -320.30612245,
       -305.40816327, -290.51020408, -275.6122449 , -260.71428571,
       -245.81632653, -230.91836735, -216.02040816, -201.12244898,
       -186.2244898 , -171.32653061, -156.42857143, -141.53061224,
       -126.63265306, -111.73469388,  -96.83673469,  -81.93877551,
        -67.04081633,  -52.14285714,  -37.24489796,  -22.34693878,
         -7.44897959,    7.44897959,   22.34693878,   37.24489796,
         52.14285714,   67.04081633,   81.93877551,   96.83673469,
        111.73469388,  126.63265306,  141.53061224,  156.42857143,
        171.32653061,  186.2244898 ,  201.12244898,  216.02040816,
        230.91836735,  245.81632653,  260.71428571,  275.6122449 ,
        290.51020408,  305.40816327,  320.30612245,  335.20408163,
        350.10204082,  365.        ])

flyby_jupiter_time_range_LIST = np.linspace(-change_in_time_days, change_in_time_days, spacing)
flyby_jupiter_time_range_LIST

array([-365.        , -350.10204082, -335.20408163, -320.30612245,
       -305.40816327, -290.51020408, -275.6122449 , -260.71428571,
       -245.81632653, -230.91836735, -216.02040816, -201.12244898,
       -186.2244898 , -171.32653061, -156.42857143, -141.53061224,
       -126.63265306, -111.73469388,  -96.83673469,  -81.93877551,
        -67.04081633,  -52.14285714,  -37.24489796,  -22.34693878,
         -7.44897959,    7.44897959,   22.34693878,   37.24489796,
         52.14285714,   67.04081633,   81.93877551,   96.83673469,
        111.73469388,  126.63265306,  141.53061224,  156.42857143,
        171.32653061,  186.2244898 ,  201.12244898,  216.02040816,
        230.91836735,  245.81632653,  260.71428571,  275.6122449 ,
        290.51020408,  305.40816327,  320.30612245,  335.20408163,
        350.10204082,  365.        ])

flyby_saturn_time_range_LIST = np.linspace(-change_in_time_days, change_in_time_days, spacing)
flyby_saturn_time_range_LIST

array([-365.        , -350.10204082, -335.20408163, -320.30612245,
       -305.40816327, -290.51020408, -275.6122449 , -260.71428571,
       -245.81632653, -230.91836735, -216.02040816, -201.12244898,
       -186.2244898 , -171.32653061, -156.42857143, -141.53061224,
       -126.63265306, -111.73469388,  -96.83673469,  -81.93877551,
        -67.04081633,  -52.14285714,  -37.24489796,  -22.34693878,
         -7.44897959,    7.44897959,   22.34693878,   37.24489796,
         52.14285714,   67.04081633,   81.93877551,   96.83673469,
        111.73469388,  126.63265306,  141.53061224,  156.42857143,
        171.32653061,  186.2244898 ,  201.12244898,  216.02040816,
        230.91836735,  245.81632653,  260.71428571,  275.6122449 ,
        290.51020408,  305.40816327,  320.30612245,  335.20408163,
        350.10204082,  365.        ])

deltaV_REAL, fuel_mass_REAL

(10.31241583834745, 1139.0)

1.3) Run Algorithm

%%time
#warnings.filterwarnings('ignore')

if global_flag:
    deltaV_list_BRUTE = []
    time_range_list_index = []
    dates_list = []
    
    # Start timing
    start_time = time.time()
    
    for i in tqdm(range(0,spacing)):
        date_departure_mjd_BRUTE = date_departure_mjd + departure_time_range_LIST[i]
        for j in range(0,spacing):
            date_flyby_jupiter_mjd_BRUTE = date_flyby_jupiter_mjd + flyby_jupiter_time_range_LIST[j]
            for k in range(0,spacing):
                date_flyby_saturn_mjd_BRUTE = date_flyby_saturn_mjd + flyby_saturn_time_range_LIST[k]
                
                time_range_list_index.append([i,j,k])
                try:
                    deltaV_mission = find_deltaV_mission(date_departure_mjd_BRUTE, date_flyby_jupiter_mjd_BRUTE, 
                                                                date_flyby_saturn_mjd_BRUTE)
                    deltaV_list_BRUTE.append(deltaV_mission)
                    dates_list.append([date_departure_mjd_BRUTE,
                                       date_flyby_jupiter_mjd_BRUTE,
                                       date_flyby_saturn_mjd_BRUTE,
                                       deltaV_mission])
                except Exception:
                    deltaV_list_BRUTE.append([1e99, 1e99 ,{'temp': 1e99}])
    
    # End timing
    end_time = time.time()
    execution_time_BRUTE_FORCE = end_time - start_time
    print(f"\nExecution Time: {execution_time_BRUTE_FORCE:.4f} seconds")

CPU times: user 3 µs, sys: 1e+03 ns, total: 4 µs
Wall time: 4.77 µs

# Contour plot of the problem space --> look at local minima
# Holo View
#deltaV_list_BRUTE
#dates_list

#deltaV_list_BRUTE

if global_flag:
    filename_brute = output_pickle.get_path("brute_results_V1.pkl")
    
    results_brute = {
        "deltaV_list_BRUTE": deltaV_list_BRUTE,
        "time_range_list_index": time_range_list_index,
        "dates_list": dates_list,
        "execution_time_BRUTE_FORCE": execution_time_BRUTE_FORCE
        
    }
    with open(filename_brute, "wb") as f:
        pickle.dump(results_brute, f)

filename_brute = output_pickle.get_path("brute_results_V1.pkl")

with open(filename_brute, "rb") as f:
    data = pickle.load(f)

deltaV_list_BRUTE = data["deltaV_list_BRUTE"]
time_range_list_index = data["time_range_list_index"]
dates_list = data["dates_list"]
execution_time_BRUTE_FORCE = data["execution_time_BRUTE_FORCE"]

if not global_flag:
    print(f"Execution Time: {execution_time_BRUTE_FORCE:.4f} seconds")

Execution Time: 412.9018 seconds

deltaV_BRUTE_MIN, fuel_mass_BRUTE_MIN, data_dict_BRUTE_MIN = min(deltaV_list_BRUTE, key=lambda x: x[1]) #min(deltaV_list_BRUTE)
deltaV_BRUTE_MIN, fuel_mass_BRUTE_MIN, data_dict_BRUTE_MIN

(9.411191461145675,
 1071.7476431066705,
 {'deltaV_injection (km/s)': 9.40484890472681,
  'deltaV_jupiter (km/s)': 0.006342556418864831,
  'rp_jupiter (km)': 820160.096156483,
  'tof (days)': 1519.1607426303601})

data_dict_BRUTE_MIN['tof (days)']

1519.1607426303601

# Find the index of the minimum value
min_index = deltaV_list_BRUTE.index(min(deltaV_list_BRUTE, key=lambda x: x[0]))
min_index

61798

deltaV_BRUTE_MIN, fuel_mass_BRUTE_MIN, data_dict_BRUTE_MIN = deltaV_list_BRUTE[min_index]
deltaV_BRUTE_MIN, fuel_mass_BRUTE_MIN, data_dict_BRUTE_MIN

(9.411191461145675,
 1071.7476431066705,
 {'deltaV_injection (km/s)': 9.40484890472681,
  'deltaV_jupiter (km/s)': 0.006342556418864831,
  'rp_jupiter (km)': 820160.096156483,
  'tof (days)': 1519.1607426303601})

change_in_time = time_range_list_index[min_index]
change_in_time

[24, 35, 48]

date_departure_mjd_BRUTE_MIN = date_departure_mjd + departure_time_range_LIST[change_in_time[0]]
date_flyby_jupiter_mjd_BRUTE_MIN = date_flyby_jupiter_mjd + flyby_jupiter_time_range_LIST[change_in_time[1]]
date_flyby_saturn_mjd_BRUTE_MIN = date_flyby_saturn_mjd + flyby_saturn_time_range_LIST[change_in_time[2]]

dates_list_BRUTE_MIN = [date_departure_mjd_BRUTE_MIN, date_flyby_jupiter_mjd_BRUTE_MIN, date_flyby_saturn_mjd_BRUTE_MIN]
dates_list_BRUTE_MIN

[43386.931575963856, 44093.93204365059, 44906.092318594216]

change_in_dates_list_BRUTE_MIN = [
    (date_departure_mjd - date_departure_mjd_BRUTE_MIN),
    (date_flyby_jupiter_mjd - date_flyby_jupiter_mjd_BRUTE_MIN),
    (date_flyby_saturn_mjd - date_flyby_saturn_mjd_BRUTE_MIN)
]
change_in_dates_list_BRUTE_MIN

[7.448979591834359, -156.42857142857247, -350.102040816324]

print("Earth-Jupiter:",date_flyby_jupiter_mjd_BRUTE_MIN - date_departure_mjd_BRUTE_MIN,"days")
print("Jupiter-Saturn:",date_flyby_saturn_mjd_BRUTE_MIN - date_flyby_jupiter_mjd_BRUTE_MIN,"days")

Earth-Jupiter: 707.000467686732 days
Jupiter-Saturn: 812.1602749436279 days

date_departure_BRUTE_MIN = MJD_to_TT_calander_date(date_departure_mjd_BRUTE_MIN)
date_flyby_jupiter_BRUTE_MIN = MJD_to_TT_calander_date(date_flyby_jupiter_mjd_BRUTE_MIN)
date_flyby_saturn_BRUTE_MIN = MJD_to_TT_calander_date(date_flyby_saturn_mjd_BRUTE_MIN)

print("Date of Departure (Min):", MJD_to_TT_calander_date(date_departure_mjd_BRUTE_MIN))

print("Date of Jupiter Flyby (Min):", MJD_to_TT_calander_date(date_flyby_jupiter_mjd_BRUTE_MIN))

print("Date of Saturn Flyby (Min):", MJD_to_TT_calander_date(date_flyby_saturn_mjd_BRUTE_MIN))

Date of Departure (Min): 1977-08-31 22:21:28.163
Date of Jupiter Flyby (Min): 1979-08-08 22:22:08.571
Date of Saturn Flyby (Min): 1981-10-29 02:12:56.327

# Extracting MJD values from the brute force results
departure_mjd = [dates_list[num][0] for num in range(0, len(dates_list))]
jupiter_flyby_mjd = [dates_list[num][1] for num in range(0, len(dates_list))]
saturn_flyby_mjd = [dates_list[num][2] for num in range(0, len(dates_list))]
deltaV_values = [dates_list[num][3][0] for num in range(0, len(dates_list))]

departure_mjd_change_in_time = [(element - date_departure_mjd) for element in departure_mjd]
jupiter_flyby_mjd_change_in_time = [(element - date_flyby_jupiter_mjd) for element in jupiter_flyby_mjd]
saturn_flyby_mjd_change_in_time = [(element - date_flyby_saturn_mjd) for element in saturn_flyby_mjd]

1.4) Contour Graph of Problem Space

# Filter out any extreme values
max_deltav_threshold = 20  # km/s (adjust as needed)
valid_indices = [i for i in range(len(deltaV_values)) if deltaV_values[i] < max_deltav_threshold]

# Extract valid data
valid_departure_mjd = [departure_mjd_change_in_time[i] for i in valid_indices]
valid_jupiter_mjd = [jupiter_flyby_mjd_change_in_time[i] for i in valid_indices]
valid_saturn_mjd = [saturn_flyby_mjd_change_in_time[i] for i in valid_indices]
valid_deltaV = [deltaV_values[i] for i in valid_indices]

# Create a 3D plot
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')



scatter = ax.scatter(valid_departure_mjd, valid_jupiter_mjd, valid_saturn_mjd, 
                     c=valid_deltaV, cmap=parula_map(True), s=5, alpha=0.25, 
                     norm=colors.Normalize(vmin=min(valid_deltaV), vmax=min(valid_deltaV) * 1.5))

# Highlight the REAL delta-V point
ax.scatter([0], 
           [0], 
           [0], 
           color='red', s=100, marker='o', edgecolors='black', linewidth=1.2, 
           zorder=3,  # Higher zorder brings it to the front
           label=f'Real Delta-V: {deltaV_REAL:.2f} km/s\nReal Fuel Mass: {fuel_mass_REAL:.2f} kg')

# Highlight the Minimum Brute Force delta-V point
ax.scatter([date_departure_mjd_BRUTE_MIN - date_departure_mjd], 
           [date_flyby_jupiter_mjd_BRUTE_MIN - date_flyby_jupiter_mjd], 
           [date_flyby_saturn_mjd_BRUTE_MIN - date_flyby_saturn_mjd], 
           color='fuchsia', s=200, marker='*', edgecolors='black', linewidth=1.2, 
           zorder=3,  # Higher zorder brings it to the front
           label=f'Minimum Delta-V: {deltaV_BRUTE_MIN:.2f} km/s\nMinimum Fuel Mass: {fuel_mass_BRUTE_MIN:.2f} kg')

cbar = plt.colorbar(scatter, ax=ax, label='$\Delta$V (km/s)')
cbar.ax.set_ylabel('$\Delta$V (km/s)', fontsize=12)

ax.set_xlabel('Change in Earth Departure Epoch (days)', labelpad=5, fontsize=12)
ax.set_ylabel('Change in Jupiter Flyby Epoch (days)', labelpad=5, fontsize=12)
ax.set_zlabel('Change in Saturn Flyby Epoch (days)', labelpad=5, fontsize=12)
ax.set_title(r"\textbf{3D Visualisation of Brute Force Problem Space ($\pm$ 365 days bounds)}", 
             fontsize=14) 

limit = 375
ax.set_xlim(-limit, limit)
ax.set_ylim(-limit, limit)
ax.set_zlim(-limit, limit)

ax.legend(loc = "best")

plt.tight_layout()
output_figures.save_file("voyager_I_brute_force_contour_3D.png", dpi=500, bbox_inches="tight")

#ax.view_init(elev=30, azim=45)  # Adjust elevation & azimuth for a better view
plt.show()

Figure saved to: Voyager 1 - Figures/voyager_I_brute_force_contour_3D.png

valid_departure_mjd = np.array(departure_mjd_change_in_time)
valid_jupiter_mjd = np.array(jupiter_flyby_mjd_change_in_time)
valid_saturn_mjd = np.array(saturn_flyby_mjd_change_in_time)
limit = 365

plt.close('all')  # Close any existing plots
fig = plt.figure(figsize=(15, 12), constrained_layout=False)
plt.subplots_adjust(top=0.9, bottom=0.05)  # Adjust top and bottom margins



ax = fig.add_subplot(111, projection='3d')

# X-Y Plane (Blue) Scatter and Surface
ax.scatter(valid_departure_mjd, valid_jupiter_mjd, -limit, color='blue', s=1, alpha=0.4)
grid_x, grid_y = np.mgrid[min(valid_departure_mjd):max(valid_departure_mjd):50j, 
                           min(valid_jupiter_mjd):max(valid_jupiter_mjd):50j]
grid_z = np.full_like(grid_x, -limit)
#ax.plot_surface(grid_x, grid_y, grid_z, color='blue', alpha=0.2)

# Y-Z Plane (Red) Scatter and Surface
ax.scatter(-limit, valid_jupiter_mjd, valid_saturn_mjd, color='red', s=1, alpha=0.4)
grid_y, grid_z = np.mgrid[min(valid_jupiter_mjd):max(valid_jupiter_mjd):50j, 
                           min(valid_saturn_mjd):max(valid_saturn_mjd):50j]
grid_x = np.full_like(grid_y, -limit)
#ax.plot_surface(grid_x, grid_y, grid_z, color='red', alpha=0.2)

# X-Z Plane (Gold) Scatter and Surface
ax.scatter(valid_departure_mjd, limit, valid_saturn_mjd, color='gold', s=1, alpha=0.4)
grid_x, grid_z = np.mgrid[min(valid_departure_mjd):max(valid_departure_mjd):50j, 
                           min(valid_saturn_mjd):max(valid_saturn_mjd):50j]
grid_y = np.full_like(grid_x, limit)
#ax.plot_surface(grid_x, grid_y, grid_z, color='gold', alpha=0.2)

ax.set_xlabel('Change in Earth Departure Epoch (days)', labelpad=3, fontsize=12)
ax.set_ylabel('Change in Jupiter Flyby Epoch (days)', labelpad=3, fontsize=12)
ax.set_zlabel('Change in Saturn Flyby Epoch (days)', labelpad=2, fontsize=12)
ax.set_title(r'\textbf{3D Visualisation of Voyager I Brute Force Problem Space ($\pm$ 365 days bounds)}', 
             fontsize=14, pad=10, y=1.02)  # Slight vertical adjustment

ax.set_xlim(-limit, limit)
ax.set_ylim(-limit, limit)
ax.set_zlim(-limit, limit)

output_figures.save_file("voyager_I_brute_force_problem_space.png", dpi=500, bbox_inches="tight", pad_inches=0.2)
plt.show()

Figure saved to: Voyager 1 - Figures/voyager_I_brute_force_problem_space.png

T_real = 0
bound = 365
sampling_list = []
K=50
DeltaT = (2*bound)/K  # 14.6 #days

for k in range(0,K+1):
    sampling_list.append( (T_real - bound) + k*(DeltaT))

len(sampling_list)

51

1.5) Fixing a Date for Problem Space 2D Contour

%%time
def optimise_trajectory(fix_parameter, spacing=50):
    
    # Validate input parameter
    if fix_parameter not in ['departure', 'jupiter', 'saturn']:
        raise ValueError("Not vaild Input.")

    time_range_LIST = np.linspace(-change_in_time_days, change_in_time_days, spacing)
    
    # Lists to store results
    deltaV_list = []
    time_range_list_index = []
    dates_list = []
    
    # Start timing
    start_time = time.time()
    
    # Determine which loops to run based on fixed parameter
    if fix_parameter == 'departure':
        # Fix departure date, vary Jupiter and Saturn
        for j in range(spacing):
            date_flyby_jupiter_mjd_BRUTE = date_flyby_jupiter_mjd + time_range_LIST[j]
            for k in range(spacing):
                date_flyby_saturn_mjd_BRUTE = date_flyby_saturn_mjd + time_range_LIST[k]
                time_range_list_index.append([0, j, k])
                try:
                    deltaV_mission = find_deltaV_mission(
                        date_departure_mjd,
                        date_flyby_jupiter_mjd_BRUTE,
                        date_flyby_saturn_mjd_BRUTE
                    )
                    deltaV_list.append(deltaV_mission)
                    dates_list.append([
                        date_departure_mjd,
                        date_flyby_jupiter_mjd_BRUTE,
                        date_flyby_saturn_mjd_BRUTE,
                        deltaV_mission
                    ])
                except Exception:
                    deltaV_list.append([1e99, 1e99, {'temp': 1e99}])
    
    elif fix_parameter == 'jupiter':
        # Fix Jupiter flyby date, vary departure and Saturn
        for i in range(spacing):
            date_departure_mjd_BRUTE = date_departure_mjd + time_range_LIST[i]
            for k in range(spacing):
                date_flyby_saturn_mjd_BRUTE = date_flyby_saturn_mjd + time_range_LIST[k]
                time_range_list_index.append([i, 0, k])
                try:
                    deltaV_mission = find_deltaV_mission(
                        date_departure_mjd_BRUTE,
                        date_flyby_jupiter_mjd,
                        date_flyby_saturn_mjd_BRUTE
                    )
                    deltaV_list.append(deltaV_mission)
                    dates_list.append([
                        date_departure_mjd_BRUTE,
                        date_flyby_jupiter_mjd,
                        date_flyby_saturn_mjd_BRUTE,
                        deltaV_mission
                    ])
                except Exception:
                    deltaV_list.append([1e99, 1e99, {'temp': 1e99}])
    
    else:  # fix_parameter == 'saturn'
        # Fix Saturn flyby date, vary departure and Jupiter
        for i in range(spacing):
            date_departure_mjd_BRUTE = date_departure_mjd + time_range_LIST[i]
            for j in range(spacing):
                date_flyby_jupiter_mjd_BRUTE = date_flyby_jupiter_mjd + time_range_LIST[j]
                time_range_list_index.append([i, j, 0])
                try:
                    deltaV_mission = find_deltaV_mission(
                        date_departure_mjd_BRUTE,
                        date_flyby_jupiter_mjd_BRUTE,
                        date_flyby_saturn_mjd
                    )
                    deltaV_list.append(deltaV_mission)
                    dates_list.append([
                        date_departure_mjd_BRUTE,
                        date_flyby_jupiter_mjd_BRUTE,
                        date_flyby_saturn_mjd,
                        deltaV_mission
                    ])
                except Exception:
                    deltaV_list.append([1e99, 1e99, {'temp': 1e99}])
    
    # End timing
    end_time = time.time()
    execution_time = end_time - start_time
    
    # Find minimum deltaV
    min_index = deltaV_list.index(min(deltaV_list, key=lambda x: x[0]))
    deltaV_BRUTE_MIN_2D, fuel_mass_BRUTE_MIN_2D, data_dict_BRUTE_MIN_2D = deltaV_list[min_index]
    
    # Get the time changes that gave minimum deltaV
    change_in_time = time_range_list_index[min_index]
    
    # Calculate the actual dates for minimum deltaV
    best_dates_BRUTE_MIN_2D = [
        date_departure_mjd + time_range_LIST[change_in_time[0]],
        date_flyby_jupiter_mjd + time_range_LIST[change_in_time[1]],
        date_flyby_saturn_mjd + time_range_LIST[change_in_time[2]]
    ]
    
    # Prepare data for plotting
    plot_data = {
        'dates_list': dates_list,
        'time_range_LIST': time_range_LIST,
        'fix_parameter': fix_parameter
    }

    return best_dates_BRUTE_MIN_2D, deltaV_BRUTE_MIN_2D, fuel_mass_BRUTE_MIN_2D, data_dict_BRUTE_MIN_2D, execution_time, plot_data

CPU times: user 6 µs, sys: 5 µs, total: 11 µs
Wall time: 17.9 µs

def plot_optimisation_results(plot_data, max_deltav_threshold=15):
    
    dates_list = plot_data['dates_list']
    fix_parameter = plot_data['fix_parameter']
    
    # Determine the x and y axis parameters based on fixed parameter
    if fix_parameter == 'departure':
        x_label = 'Change in Jupiter Flyby Epoch (days)'
        y_label = 'Change in Saturn Flyby Epoch (days)'
        title = r'\textbf{2D Contour Graph of Problem Space with Earth Departure Fixed ($\pm$365 days)}'
        x_mjd_ref = date_flyby_jupiter_mjd
        y_mjd_ref = date_flyby_saturn_mjd
        x_index = 1
        y_index = 2
    elif fix_parameter == 'jupiter':
        x_label = 'Change in Earth Departure Epoch (days)'
        y_label = 'Change in Saturn Flyby Epoch (days)'
        title = r'\textbf{2D Contour Graph of Problem Space with Jupiter Flyby Fixed ($\pm$365 days)}'
        x_mjd_ref = date_departure_mjd
        y_mjd_ref = date_flyby_saturn_mjd
        x_index = 0
        y_index = 2
    else:  # fix_parameter == 'saturn'
        x_label = 'Change in Earth Departure Epoch (days)'
        y_label = 'Change in Jupiter Flyby Epoch (days)'
        title = r'\textbf{2D Contour Graph of Problem Space with Saturn Flyby Fixed ($\pm$365 days)}'
        x_mjd_ref = date_departure_mjd
        y_mjd_ref = date_flyby_jupiter_mjd
        x_index = 0
        y_index = 1
    
    # Extract x, y, and deltaV values
    x_mjd = [dates_list[num][x_index] for num in range(len(dates_list))]
    y_mjd = [dates_list[num][y_index] for num in range(len(dates_list))]
    deltaV_values = [dates_list[num][3][0] for num in range(len(dates_list))]
    
    # Convert to change in days
    x_mjd_change = [(element - x_mjd_ref) for element in x_mjd]
    y_mjd_change = [(element - y_mjd_ref) for element in y_mjd]
    
    # Filter out invalid values (too high deltaV)
    valid_indices = [i for i in range(len(deltaV_values)) if deltaV_values[i] < max_deltav_threshold]
    
    # Extract valid data
    valid_x_mjd = [x_mjd_change[i] for i in valid_indices]
    valid_y_mjd = [y_mjd_change[i] for i in valid_indices]
    valid_deltaV = [deltaV_values[i] for i in valid_indices]
    
    # Grid sizes for interpolation
    x_range = np.linspace(min(valid_x_mjd), max(valid_x_mjd), 1000)
    y_range = np.linspace(min(valid_y_mjd), max(valid_y_mjd), 1000)
    
    # Create meshgrid
    X, Y = np.meshgrid(x_range, y_range)
    
    # Interpolate deltaV values to the grid using linear interpolation
    Z = griddata((valid_x_mjd, valid_y_mjd), valid_deltaV, (X, Y), method='linear')
    
    # Create the contour plot
    fig, ax = plt.subplots(figsize=(10, 8))
    
    plt.rc('text', usetex=True)
    plt.rc('font', family='serif')
    
    cmap = parula_map(True)
    
    # Define contour levels within the specified range
    min_deltav = np.nanmin(Z)
    levels = np.linspace(min_deltav, max_deltav_threshold, 20)
    
    # Create filled contour plot
    contour = ax.contourf(X, Y, Z, levels=levels, cmap=cmap, 
                          norm=colors.Normalize(vmin=min_deltav, vmax=max_deltav_threshold))
    
    # Add colorbar
    cbar = plt.colorbar(contour, ax=ax, label=r'$\Delta V$ (km/s)')
    cbar.ax.tick_params(labelsize=12)
    
    # Add contour lines
    contour_lines = ax.contour(X, Y, Z, levels=10, colors='white', linewidths=0.5, alpha=0.7)
    ax.clabel(contour_lines, inline=True, fontsize=8, fmt='%.2f')
    
    # Find minimum deltaV point
    min_idx = valid_deltaV.index(min(valid_deltaV))
    min_x = valid_x_mjd[min_idx]
    min_y = valid_y_mjd[min_idx]
    min_dv = valid_deltaV[min_idx]
    
    # Plot original and minimum deltaV points
    ax.scatter(0, 0, color='red', s=100, marker='o', edgecolors='black', linewidth=1.2, 
           label=r'Real $\Delta V$: {:.2f} km/s'.format(deltaV_REAL))
    
    #ax.scatter(min_x, min_y, color='fuchsia', s=200, marker='*', edgecolors='black', linewidth=1.2,
    #          label=f'Minimum $\Delta V$: {min_dv:.2f} km/s')
    
    ax.set_xlabel(x_label, labelpad=2, fontsize=12)
    ax.set_ylabel(y_label, labelpad=2, fontsize=12)
    ax.set_title(title, pad=15, fontsize=14)
    
    ax.minorticks_on()
    ax.tick_params(axis='both', which='major', labelsize=12)
    ax.tick_params(axis='both', which='minor', labelsize=10)

    #plt.xlim(-365,365)
    #plt.ylim(-365,365)
    
    plt.legend(loc='upper left')
    
    plt.tight_layout()

    output_figures.save_file(f"voyager_1_brute_force_{fix_parameter}_fixed_2D.png", dpi=500, bbox_inches="tight")

    plt.show()

fix_parameter="departure"

BRUTE_MIN_2D_fixed_departure = optimise_trajectory("departure", 50)
BRUTE_MIN_2D_fixed_jupiter = optimise_trajectory("jupiter", 50)
BRUTE_MIN_2D_fixed_saturn = optimise_trajectory("saturn", 50)

plot_optimisation_results(BRUTE_MIN_2D_fixed_departure[-1], 15)
plot_optimisation_results(BRUTE_MIN_2D_fixed_jupiter[-1], 15)
plot_optimisation_results(BRUTE_MIN_2D_fixed_saturn[-1], 15)

/var/folders/ts/2k06pkqj7sqgrljsl3sdbxc80000gn/T/ipykernel_55087/4231714446.py:20: RuntimeWarning: overflow encountered in cosh
  return (np.cosh(np.sqrt(-z)) - 1) / (-z)
/var/folders/ts/2k06pkqj7sqgrljsl3sdbxc80000gn/T/ipykernel_55087/4231714446.py:29: RuntimeWarning: overflow encountered in sinh
  return (np.sinh(np.sqrt(-z)) - np.sqrt(-z)) / ((-z) ** 1.5)
/var/folders/ts/2k06pkqj7sqgrljsl3sdbxc80000gn/T/ipykernel_55087/4231714446.py:37: RuntimeWarning: invalid value encountered in scalar divide
  return max(r1 + r2 + A * (z * S(z) - 1) / np.sqrt(C_z), 1e-8)

Figure saved to: Voyager 1 - Figures/voyager_1_brute_force_departure_fixed_2D.png

Figure saved to: Voyager 1 - Figures/voyager_1_brute_force_jupiter_fixed_2D.png

Figure saved to: Voyager 1 - Figures/voyager_1_brute_force_saturn_fixed_2D.png

---

Fmin Numerical Method

https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin.html

https://jasoncantarella.com/downloads/NelderMeadProof.pdf

dates_REAL

[43394.38055555569, 43937.503472222015, 44555.99027777789]

# Initialise an empty list to store the iterations
iterations = []

# Initial guess
initial_guess = np.array([date_departure_mjd, date_flyby_jupiter_mjd, date_flyby_saturn_mjd])

def objective_function(dates):
    # Ensure dates stay within ±365 days of the initial guess
    date_departure_mjd, date_flyby_jupiter_mjd, date_flyby_saturn_mjd = dates

    # Ensure dates stay within ±365 days of the initial guess
    lower_bound = initial_guess - 365
    upper_bound = initial_guess + 365
    
    # Check if any value is outside bounds
    if np.any(dates < lower_bound) or np.any(dates > upper_bound):
        deltaV = 1e99
    else:
        try:
            deltaV, fuel_mass, data_dict = find_deltaV_mission(date_departure_mjd, date_flyby_jupiter_mjd, date_flyby_saturn_mjd)
            
            # Store the current iteration (dates)
            iterations.append(dates.copy())
        except Exception:
            deltaV = 1e99  # Assign a high penalty for invalid solutions
        
    return deltaV  # Minimise Delta-V

# Start timing
start_time = time.time()

# Run optimisation using `fmin` with manual bounds enforcement
optimal_dates = fmin(objective_function, initial_guess, disp=True)

# Extract optimised dates
date_departure_mjd_FMIN, date_flyby_jupiter_mjd_FMIN, date_flyby_saturn_mjd_FMIN = optimal_dates

# Compute Delta-V and fuel mass at optimised dates
deltaV_FMIN, fuel_mass_FMIN, data_dict_FMIN = find_deltaV_mission(date_departure_mjd_FMIN, date_flyby_jupiter_mjd_FMIN, date_flyby_saturn_mjd_FMIN)

# Display results
print(f"Minimum Delta-V: {deltaV_FMIN:.4f} km/s")
print(f"Optimal Departure Date (MJD): {date_departure_mjd_FMIN:.2f}")
print(f"Optimal Jupiter Flyby Date (MJD): {date_flyby_jupiter_mjd_FMIN:.2f}")
print(f"Optimal Saturn Flyby Date (MJD): {date_flyby_saturn_mjd_FMIN:.2f}")

# Convert to calendar dates
print("\nIn calendar dates:")
print(f"Departure: {MJD_to_TT_calander_date(date_departure_mjd_FMIN)}")
print(f"Jupiter flyby: {MJD_to_TT_calander_date(date_flyby_jupiter_mjd_FMIN)}")
print(f"Saturn flyby: {MJD_to_TT_calander_date(date_flyby_saturn_mjd_FMIN)}")

# End timing
end_time = time.time()
execution_time_FMIN = end_time - start_time
print(f"\nExecution Time: {execution_time_FMIN:.4f} seconds")

Optimization terminated successfully.
         Current function value: 9.388030
         Iterations: 201
         Function evaluations: 363
Minimum Delta-V: 9.3880 km/s
Optimal Departure Date (MJD): 43388.37
Optimal Jupiter Flyby Date (MJD): 44099.45
Optimal Saturn Flyby Date (MJD): 44920.99

In calendar dates:
Departure: 1977-09-02 08:54:49.416
Jupiter flyby: 1979-08-14 10:52:57.591
Saturn flyby: 1981-11-12 23:45:59.999

Execution Time: 1.1450 seconds

#iterations

deltaV_FMIN, fuel_mass_FMIN, data_dict_FMIN = find_deltaV_mission(date_departure_mjd_FMIN, date_flyby_jupiter_mjd_FMIN, date_flyby_saturn_mjd_FMIN)
deltaV_FMIN, fuel_mass_FMIN, data_dict_FMIN

(9.388030073904538,
 1069.9564197303307,
 {'deltaV_injection (km/s)': 9.388030073866002,
  'deltaV_jupiter (km/s)': 3.8536285273949034e-11,
  'rp_jupiter (km)': 848168.1721327668,
  'tof (days)': 1532.6188724847307})

print("Earth-Jupiter:",date_flyby_jupiter_mjd_FMIN - date_departure_mjd_FMIN,"days")
print("Jupiter-Saturn:",date_flyby_saturn_mjd_FMIN - date_flyby_jupiter_mjd_FMIN,"days")

Earth-Jupiter: 711.0820390666777 days
Jupiter-Saturn: 821.536833418053 days

dates_list_FMIN = [date_departure_mjd_FMIN, date_flyby_jupiter_mjd_FMIN, date_flyby_saturn_mjd_FMIN]
dates_list_FMIN

[43388.371405276324, 44099.453444343, 44920.990277761055]

change_in_dates_list_FMIN = [
    (date_departure_mjd - date_departure_mjd_FMIN),
    (date_flyby_jupiter_mjd - date_flyby_jupiter_mjd_FMIN),
    (date_flyby_saturn_mjd - date_flyby_saturn_mjd_FMIN)
]
change_in_dates_list_FMIN

[6.009150279365713, -161.94997212098679, -364.99999998316343]

date_departure_FMIN = MJD_to_TT_calander_date(date_departure_mjd_FMIN)
date_flyby_jupiter_FMIN = MJD_to_TT_calander_date(date_flyby_jupiter_mjd_FMIN)
date_flyby_saturn_FMIN = MJD_to_TT_calander_date(date_flyby_saturn_mjd_FMIN)

print("Date of Departure (Min):", MJD_to_TT_calander_date(date_departure_mjd_FMIN))

print("Date of Jupiter Flyby (Min):", MJD_to_TT_calander_date(date_flyby_jupiter_mjd_FMIN))

print("Date of Saturn Flyby (Min):", MJD_to_TT_calander_date(date_flyby_saturn_mjd_FMIN))

Date of Departure (Min): 1977-09-02 08:54:49.416
Date of Jupiter Flyby (Min): 1979-08-14 10:52:57.591
Date of Saturn Flyby (Min): 1981-11-12 23:45:59.999

Contour Graph of Problem Space

# Extract the dates from the iterations list
departure_dates_FMIN = np.array([iter[0] for iter in iterations])
jupiter_flyby_dates_FMIN = np.array([iter[1] for iter in iterations])
saturn_flyby_dates_FMIN = np.array([iter[2] for iter in iterations])

# Calculate the corresponding deltaV values for the iterations
deltaV_values_FMIN = []
for dates in iterations:
    try:
        deltaV, _, _ = find_deltaV_mission(*dates)
    except Exception:
        deltaV_values_FMIN.append(1e99)
        continue
    deltaV_values_FMIN.append(deltaV)

# Create a 3D plot
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')



# Scatter plot with color based on deltaV
scatter = ax.scatter(departure_dates_FMIN - date_departure_mjd, jupiter_flyby_dates_FMIN - date_flyby_jupiter_mjd, saturn_flyby_dates_FMIN - date_flyby_saturn_mjd, 
                     c=deltaV_values_FMIN, cmap=parula_map(True), s=10, alpha=0.7, 
                     norm=plt.Normalize(vmin=min(deltaV_values_FMIN), vmax=min(deltaV_values_FMIN) * 1.5))

#ax.plot(departure_dates_FMIN - date_departure_mjd, jupiter_flyby_dates_FMIN - date_flyby_jupiter_mjd, saturn_flyby_dates_FMIN - date_flyby_saturn_mjd,
#        linestyle='-', color='g', markersize=6, linewidth=0.5, alpha=0.5)

ax.scatter([0], 
           [0], 
           [0], 
           color='red', s=100, marker='o', edgecolors='black', linewidth=1.2, 
           zorder=3,  # Higher zorder brings it to the front
           label=f'Real Delta-V: {deltaV_REAL:.2f} km/s\nReal Fuel Mass: {fuel_mass_REAL:.2f} kg')

ax.scatter(date_departure_mjd_FMIN - date_departure_mjd, 
           date_flyby_jupiter_mjd_FMIN - date_flyby_jupiter_mjd, 
           date_flyby_saturn_mjd_FMIN - date_flyby_saturn_mjd, 
           color='fuchsia', s=200, marker='*', edgecolors='black', linewidth=1.2, 
           zorder=3,  # Higher zorder brings it to the front
           label=f'Minimum Delta-V: {deltaV_FMIN:.4f} km/s\nMinimum Fuel Mass: {fuel_mass_FMIN:.2f} kg')

cbar = plt.colorbar(scatter, ax=ax, label='$\Delta$V (km/s)')
cbar.ax.set_ylabel('$\Delta$V (km/s)', fontsize=12)

ax.set_xlabel('Change in Earth Departure Epoch (days)', fontsize=12)
ax.set_ylabel('Change in Jupiter Flyby Epoch (days)', fontsize=12)
ax.set_zlabel('Change in Saturn Flyby Epoch (days)', fontsize=12)
ax.set_title(r'\textbf{3D Visualisation of Fmin Problem Space ($\pm$ 365 days bounds)}', fontsize=14)

ax.legend(loc = "best")

# Set axis limits
# limit = 375
# ax.set_xlim(-limit, limit)
# ax.set_ylim(-limit, limit)
# ax.set_zlim(-limit, limit)

# Show the plot
plt.tight_layout()

output_figures.save_file("voyager_I_fmin_contour_3D.png", dpi=500, bbox_inches="tight")

plt.show()

Figure saved to: Voyager 1 - Figures/voyager_I_fmin_contour_3D.png

#find_deltaV_mission(departure_dates[-1], jupiter_flyby_dates[-1], saturn_flyby_dates[-1])
change_in_dates_list_FMIN

[6.009150279365713, -161.94997212098679, -364.99999998316343]

---

Genetic Algorithm (GA)

min_departure_date = date_departure_mjd - change_in_time_days 
max_departure_date = date_departure_mjd + change_in_time_days

min_flyby_jupiter = date_flyby_jupiter_mjd - change_in_time_days
max_flyby_jupiter = date_flyby_jupiter_mjd + change_in_time_days

min_flyby_saturn = date_flyby_saturn_mjd - change_in_time_days 
max_flyby_saturn = date_flyby_saturn_mjd + change_in_time_days  

Step 1: Generate Initial Population

def initial_population(n_population):
    # Generates initial population within mission constraints.
    population = []
    population.append(dates_REAL)

    # Add interval to generate population
    for num in range(n_population-1):
        date_departure_pop = random.uniform(min_departure_date, max_departure_date)
        date_flyby_jupiter_pop = random.uniform(min_flyby_jupiter, max_flyby_jupiter)
        date_flyby_saturn_pop = random.uniform(min_flyby_saturn, max_flyby_saturn)

        # Ensure chronological order of mission events
        date_flyby_jupiter_pop = max(date_flyby_jupiter_pop, date_departure_pop)
        date_flyby_saturn_pop = max(date_flyby_saturn_pop, date_flyby_jupiter_pop)

        population.append([date_departure_pop, date_flyby_jupiter_pop, date_flyby_saturn_pop])
    
    return population

#pop = initial_population(2000)

Step 2: Evaluation [Fitness Function (Objective Function)]

def fitness(individual):
    # Evaluates fitness based on deltaV_total minimisation.
    try:
        date_departure_mjd, date_flyby_jupiter_mjd, date_flyby_saturn_mjd = individual
        deltaV, fuel_mass, data_dict = find_deltaV_mission(*individual)
    except Exception:
        deltaV = 1e6 #np.inf  # Assign a large penalty for infeasible solutions

    return 1 / (deltaV + 1e-6)  # Avoid division by zero

#np.array([fitness(ind) for ind in pop])

Step 3: Sort Population by Fitness

def sort_population_by_fitness(population, fitness_scores, n_population):
    sorted_indices = np.argsort(fitness_scores)[::-1]  # Sort by highest fitness
    population = [population[i] for i in sorted_indices[:n_population]] # Ensures length population is 'n_population' 
    return population   

Step 4: Elitism

https://algorithmafternoon.com/genetic/elitist_genetic_algorithm/

def elitism(population, n_elite):
    elite_individuals = population[:n_elite]
    return elite_individuals

Step 5: Selection

Tournament Selection

def tournament_selection(population, fitness_scores, tournament_size=3):
    # Select random individuals for the tournament
    selected_indices = np.random.choice(len(population), tournament_size, replace=False)
    
    # Find the best individual in the tournament
    tournament_fitness = [fitness_scores[i] for i in selected_indices]
    winner_idx = selected_indices[np.argmax(tournament_fitness)]
    
    return population[winner_idx]

Roulette Wheel Selection

def roulette_wheel_selection(population, fitness_values):
    # Calculate the normalised selection probability for each individual
    selection_prob = fitness_values / sum(fitness_values)
    # Calculate cumulative sum of fitness probabilities
    cum_sum = selection_prob.cumsum()
    # Generate a random number between 0 and 1
    r = random.random()
    # Find the first index where cumulative sum exceeds random number
    selected_index = np.where(cum_sum >= r)[0][0]
    return population[selected_index]

pop = initial_population(20)
# Evaluate fitness function
fitness_scores = np.array([fitness(ind) for ind in pop])        
# Sorts population by fitness value
pop = sort_population_by_fitness(pop, fitness_scores, 20)

roulette_wheel_selection(pop, fitness_scores)

[43681.45264305911, 43878.22580306475, 44672.21256580331]

Step 6: Crossover

Single-Point Crossover

def single_point_crossover(parent_1, parent_2):
    #Performs single-point crossover.
    crossover_point = random.randint(1, 2)
    offspring_1 = parent_1[:crossover_point] + parent_2[crossover_point:]
    offspring_2 = parent_2[:crossover_point] + parent_1[crossover_point:]

    # Ensure chronological order
    offspring_1[1] = max(offspring_1[1], offspring_1[0])
    offspring_1[2] = max(offspring_1[2], offspring_1[1])
    offspring_2[1] = max(offspring_2[1], offspring_2[0])
    offspring_2[2] = max(offspring_2[2], offspring_2[1])

    return offspring_1, offspring_2

Simulated Binary Crossover (SBX)

https://pymoo.org/operators/crossover.html

def sbx_crossover(parent_1, parent_2, eta=1.5):
    offspring_1 = parent_1[:]
    offspring_2 = parent_2[:]
    
    for i in range(len(parent_1)):  
        u = random.random()
        if u <= 0.5:
            beta = (2 * u) ** (1 / (eta + 1))
        else:
            beta = (1 / (2 * (1 - u))) ** (1 / (eta + 1))

        # Generate offspring
        offspring_1[i] = 0.5 * ((1 + beta) * parent_1[i] + (1 - beta) * parent_2[i])
        offspring_2[i] = 0.5 * ((1 - beta) * parent_1[i] + (1 + beta) * parent_2[i])

    # Ensure chronological order (if values represent time-based sequences)
    offspring_1[0] = min(offspring_1[1], offspring_1[0])
    offspring_1[1] = max(offspring_1[1], offspring_1[0])
    offspring_1[2] = max(offspring_1[2], offspring_1[1])
    
    offspring_2[0] = min(offspring_2[1], offspring_2[0])
    offspring_2[1] = max(offspring_2[1], offspring_2[0])
    offspring_2[2] = max(offspring_2[2], offspring_2[1])

    return offspring_1, offspring_2

Step 7: Mutation (Random Perturbations)

def uniform_mutation(individual):
    # Create a copy to avoid modifying the original
    mutated = individual.copy()
    
    # Define mutation strength
    mutation_strength = 30  # Days to perturb randomly
    
    # Select random index to mutate
    index = random.randint(0, 2)
    
    # Apply uniform mutation
    mutated[index] += random.uniform(-mutation_strength, mutation_strength)
    
    # Ensure sequential order
    mutated[0] = min(mutated[1], mutated[0])
    mutated[1] = max(mutated[1], mutated[0])
    mutated[2] = max(mutated[2], mutated[1])
    
    return mutated

def gaussian_mutation(individual):
    # Create a copy to avoid modifying the original
    mutated = individual.copy()
    
    # Define bounds for each parameter
    bounds = [
        (min_departure_date, max_departure_date),
        (min_flyby_jupiter, max_flyby_jupiter),
        (min_flyby_saturn, max_flyby_saturn)
    ]
    
    # Select random index to mutate
    index = random.randint(0, 2)
    
    # Calculate parameter range
    param_range = bounds[index][1] - bounds[index][0]
    
    # Apply Gaussian mutation with standard deviation proportional to parameter range
    sigma = 0.05 * param_range  # 5% of the parameter range
    delta = random.gauss(0, sigma)
    mutated[index] += delta
    
    # Ensure value stays within bounds
    mutated[index] = max(min(mutated[index], bounds[index][1]), bounds[index][0])
    
    # Ensure sequential order
    mutated[0] = min(mutated[1], mutated[0])
    mutated[1] = max(mutated[1], mutated[0])
    mutated[2] = max(mutated[2], mutated[1])
    
    return mutated

Step 8: Running GA

1. Initialise population
2. Sort population
3. Start Loop
4. Evaluate fitness
5. Select parents (roulette wheel)
6. Crossover (SBX or copy)
7. Mutation
8. Combine Offspring & Elite Individuals
9. Filter for constraints
10. Sort & truncate population
11. Elitism (store best individuals)
12. Does it satisfy termination condition
13. If NO, then repeat loop. If YES, end loop
14. Store best (optimal) individual solution

def run_genetic_algorithm(n_population, n_elite, crossover_per, mutation_per, n_generations, population_per_generation):
    #n_generations = n_generations + 1
    population = initial_population(n_population) # Generate initial population

    # Evaluate fitness function
    fitness_scores_temp = np.array([fitness(ind) for ind in population])
    # Sorts population by fitness value
    population_temp = sort_population_by_fitness(population, fitness_scores_temp, n_population)
    
    population_per_generation.append(population_temp)
    elite_individuals = []
    
    for generation in range(0, n_generations):
        start_time = time.time()
        # Evaluate fitness function
        fitness_scores = np.array([fitness(ind) for ind in population])
            
        # Sorts population by fitness value
        population = sort_population_by_fitness(population, fitness_scores, n_population)

        # Print progress
        best_individual = population[0]
        best_deltaV, best_fuel_mass, _ = find_deltaV_mission(*best_individual)
        print(f"Generation {generation}: Best Delta-V = {best_deltaV:.6f} km/s, Fitness = {fitness(best_individual):.6f}")
                
        # Apply elitism - preserve the best individuals
        if n_elite != 0:
            elite_individuals = elitism(population, n_elite)
        
        # Select parents for potential crossover - select enough for the whole population
        parents_list = []
        for i in range(n_population):
            parents_list.append(tournament_selection(population, fitness_scores))

        # Perform crossover with crossover probability
        offspring_list = []
        for i in range(0, len(parents_list) - 1, 2):
            # Apply crossover based on crossover probability
            if random.random() < crossover_per:
                offspring_1, offspring_2 = sbx_crossover(parents_list[i], parents_list[i+1])
            else:
                # If no crossover, children are exact copies of parents
                offspring_1, offspring_2 = parents_list[i][:], parents_list[i+1][:]
            
            offspring_list.extend([offspring_1, offspring_2])

        # Perform mutation
        for i in range(len(offspring_list)):
            if random.random() < mutation_per:
                    offspring_list[i] = gaussian_mutation(offspring_list[i])
        
        # Combine elite individuals, parents and offspring
        mixed_population = offspring_list + elite_individuals
        
        # Filter solutions that meet the mission constraints
        mixed_population = [
            ind for ind in mixed_population
            if np.abs(ind[0] - date_departure_mjd) <= change_in_time_days
            and np.abs(ind[1] - date_flyby_jupiter_mjd) <= change_in_time_days
            and np.abs(ind[2] - date_flyby_saturn_mjd) <= change_in_time_days
        ]
        
        # Ensure population is not empty (if all individuals were filtered out)
        if not mixed_population:
            mixed_population = offspring_list + elite_individuals # Restore full population if filtering removed all

        # Ensure population size remains constant (selection 'n_population' best individuals)
        fitness_scores = np.array([fitness(ind) for ind in mixed_population])
        population = sort_population_by_fitness(mixed_population, fitness_scores, n_population)

        # Store current population
        population_per_generation.append(population)

        end_time = time.time()
        #print(end_time - start_time)
    
    # Return the best-found solution
    best_individual = population[0]
    best_list = best_deltaV, best_fuel_mass, best_data = find_deltaV_mission(*best_individual)
    print(f"Generation {n_generations}: Best Delta-V = {best_deltaV:.6f} km/s, Fitness = {fitness(best_individual):.6f}")
    
    print("\nFinal Optimised Solution:")
    print(f"Departure Date (MJD): {best_individual[0]:.2f}")
    print(f"Jupiter Flyby Date (MJD): {best_individual[1]:.2f}")
    print(f"Saturn Flyby Date (MJD): {best_individual[2]:.2f}")
    print(f"Total deltaV: {best_deltaV:.6f} km/s")
    
    return best_individual, best_list, population_per_generation

# n_population = 2000
# n_elite = 10
# crossover_per = 0.9
# mutation_per = 0.3
# n_generations = 20

n_population = 1500
n_elite = 10
crossover_per = 0.9
mutation_per = 0.35
n_generations = 50

# n_population = 300
# n_elite = 10
# crossover_per = 0.9
# mutation_per = 0.35
# n_generations = 50

%%time

if global_flag:
    
    # Start timing
    start_time = time.time()
    
    # Run the Genetic Algorithm
    best_individual, best_mission_data, population_per_generation = run_genetic_algorithm(n_population, n_elite, crossover_per, mutation_per, n_generations, [])
    
    # End timing
    end_time = time.time()
    
    execution_time_GA = end_time - start_time
    
    print(f"\nExecution Time: {execution_time_GA:.4f} seconds\n")

CPU times: user 3 µs, sys: 5 µs, total: 8 µs
Wall time: 42 µs

if global_flag:
    filename_GA = output_pickle.get_path("ga_results_V1.pkl")
    
    results_GA = {
        "Best Individual": best_individual,
        "Best Mission Data": best_mission_data,
        "Population Per Generation": population_per_generation,
        "execution_time_GA": execution_time_GA
    }
    
    with open(filename_GA, "wb") as f:
        pickle.dump(results_GA, f)

filename_GA = output_pickle.get_path("ga_results_V1.pkl")

with open(filename_GA, "rb") as f:
    data = pickle.load(f)

best_individual = data["Best Individual"]
best_mission_data = data["Best Mission Data"]
population_per_generation = data["Population Per Generation"]
execution_time_GA = data["execution_time_GA"]

if not global_flag:
    for gen in range(len(population_per_generation)):
        best_individual = population_per_generation[gen][0]
        best_deltaV, best_fuel_mass, _ = find_deltaV_mission(*best_individual)
        print(f"Generation {gen}: Best Delta-V = {best_deltaV:.6f} km/s, Fitness = {fitness(best_individual):.6f}")

    # Return the best-found solution
    best_individual = population_per_generation[-1][0]
    best_list = best_deltaV, best_fuel_mass, best_data = find_deltaV_mission(*best_individual)
    print(f"Generation {n_generations}: Best Delta-V = {best_deltaV:.6f} km/s, Fitness = {fitness(best_individual):.6f}")
    
    print("\nFinal Optimised Solution:")
    print(f"Departure Date (MJD): {best_individual[0]:.2f}")
    print(f"Jupiter Flyby Date (MJD): {best_individual[1]:.2f}")
    print(f"Saturn Flyby Date (MJD): {best_individual[2]:.2f}")
    print(f"Total deltaV: {best_deltaV:.6f} km/s")

    print(f"\nExecution Time: {execution_time_GA:.4f} seconds\n")

Generation 0: Best Delta-V = 9.673398 km/s, Fitness = 0.103376
Generation 1: Best Delta-V = 9.673398 km/s, Fitness = 0.103376
Generation 2: Best Delta-V = 9.673398 km/s, Fitness = 0.103376
Generation 3: Best Delta-V = 9.658887 km/s, Fitness = 0.103532
Generation 4: Best Delta-V = 9.541725 km/s, Fitness = 0.104803
Generation 5: Best Delta-V = 9.458555 km/s, Fitness = 0.105724
Generation 6: Best Delta-V = 9.423885 km/s, Fitness = 0.106113
Generation 7: Best Delta-V = 9.423885 km/s, Fitness = 0.106113
Generation 8: Best Delta-V = 9.414176 km/s, Fitness = 0.106223
Generation 9: Best Delta-V = 9.392526 km/s, Fitness = 0.106468
Generation 10: Best Delta-V = 9.392526 km/s, Fitness = 0.106468
Generation 11: Best Delta-V = 9.392526 km/s, Fitness = 0.106468
Generation 12: Best Delta-V = 9.389653 km/s, Fitness = 0.106500
Generation 13: Best Delta-V = 9.389653 km/s, Fitness = 0.106500
Generation 14: Best Delta-V = 9.389653 km/s, Fitness = 0.106500
Generation 15: Best Delta-V = 9.388551 km/s, Fitness = 0.106513
Generation 16: Best Delta-V = 9.388551 km/s, Fitness = 0.106513
Generation 17: Best Delta-V = 9.388551 km/s, Fitness = 0.106513
Generation 18: Best Delta-V = 9.388551 km/s, Fitness = 0.106513
Generation 19: Best Delta-V = 9.388331 km/s, Fitness = 0.106515
Generation 20: Best Delta-V = 9.388191 km/s, Fitness = 0.106517
Generation 21: Best Delta-V = 9.388191 km/s, Fitness = 0.106517
Generation 22: Best Delta-V = 9.388191 km/s, Fitness = 0.106517
Generation 23: Best Delta-V = 9.388191 km/s, Fitness = 0.106517
Generation 24: Best Delta-V = 9.388140 km/s, Fitness = 0.106517
Generation 25: Best Delta-V = 9.388060 km/s, Fitness = 0.106518
Generation 26: Best Delta-V = 9.388060 km/s, Fitness = 0.106518
Generation 27: Best Delta-V = 9.388060 km/s, Fitness = 0.106518
Generation 28: Best Delta-V = 9.388042 km/s, Fitness = 0.106518
Generation 29: Best Delta-V = 9.388042 km/s, Fitness = 0.106518
Generation 30: Best Delta-V = 9.388042 km/s, Fitness = 0.106518
Generation 31: Best Delta-V = 9.388042 km/s, Fitness = 0.106518
Generation 32: Best Delta-V = 9.388042 km/s, Fitness = 0.106518
Generation 33: Best Delta-V = 9.388042 km/s, Fitness = 0.106518
Generation 34: Best Delta-V = 9.388042 km/s, Fitness = 0.106518
Generation 35: Best Delta-V = 9.388039 km/s, Fitness = 0.106519
Generation 36: Best Delta-V = 9.388038 km/s, Fitness = 0.106519
Generation 37: Best Delta-V = 9.388033 km/s, Fitness = 0.106519
Generation 38: Best Delta-V = 9.388031 km/s, Fitness = 0.106519
Generation 39: Best Delta-V = 9.388031 km/s, Fitness = 0.106519
Generation 40: Best Delta-V = 9.388031 km/s, Fitness = 0.106519
Generation 41: Best Delta-V = 9.388031 km/s, Fitness = 0.106519
Generation 42: Best Delta-V = 9.388031 km/s, Fitness = 0.106519
Generation 43: Best Delta-V = 9.388031 km/s, Fitness = 0.106519
Generation 44: Best Delta-V = 9.388030 km/s, Fitness = 0.106519
Generation 45: Best Delta-V = 9.388030 km/s, Fitness = 0.106519
Generation 46: Best Delta-V = 9.388030 km/s, Fitness = 0.106519
Generation 47: Best Delta-V = 9.388030 km/s, Fitness = 0.106519
Generation 48: Best Delta-V = 9.388030 km/s, Fitness = 0.106519
Generation 49: Best Delta-V = 9.388030 km/s, Fitness = 0.106519
Generation 50: Best Delta-V = 9.388030 km/s, Fitness = 0.106519
Generation 50: Best Delta-V = 9.388030 km/s, Fitness = 0.106519

Final Optimised Solution:
Departure Date (MJD): 43388.37
Jupiter Flyby Date (MJD): 44099.45
Saturn Flyby Date (MJD): 44920.99
Total deltaV: 9.388030 km/s

Execution Time: 363.4540 seconds

best_solutions_per_generation = [element[0] for element in population_per_generation]
best_solutions_per_generation, len(best_solutions_per_generation)

([[43395.24936125339, 44092.91517706043, 44880.85431078622],
  [43395.24936125339, 44092.91517706043, 44880.85431078622],
  [43395.24936125339, 44092.91517706043, 44880.85431078622],
  [43387.26530285188, 44031.50633424819, 44750.20799249409],
  [43383.92821875731, 44093.27800066935, 44889.28658652461],
  [43386.59838086187, 44086.44605020808, 44881.147748071206],
  [43390.383227149854, 44095.412559876946, 44913.24275187008],
  [43390.383227149854, 44095.412559876946, 44913.24275187008],
  [43388.18242637586, 44087.54321989983, 44892.05273861021],
  [43388.16004720267, 44099.95270596447, 44920.99027777789],
  [43388.16004720267, 44099.95270596447, 44920.99027777789],
  [43388.16004720267, 44099.95270596447, 44920.99027777789],
  [43388.43621712814, 44099.65258156039, 44920.99027777789],
  [43388.43621712814, 44099.65258156039, 44920.99027777789],
  [43388.43621712814, 44099.65258156039, 44920.99027777789],
  [43388.5069544984, 44099.428223188144, 44920.99027777789],
  [43388.5069544984, 44099.428223188144, 44920.99027777789],
  [43388.5069544984, 44099.428223188144, 44920.99027777789],
  [43388.5069544984, 44099.428223188144, 44920.99027777789],
  [43388.33500470211, 44099.36300614009, 44920.80595897071],
  [43388.21305093977, 44099.403797156214, 44920.90103293918],
  [43388.21305093977, 44099.403797156214, 44920.90103293918],
  [43388.21305093977, 44099.403797156214, 44920.90103293918],
  [43388.21305093977, 44099.403797156214, 44920.90103293918],
  [43388.279109734205, 44099.438543158794, 44920.99027777789],
  [43388.30055327698, 44099.448358952395, 44920.99027777789],
  [43388.30055327698, 44099.448358952395, 44920.99027777789],
  [43388.30055327698, 44099.448358952395, 44920.99027777789],
  [43388.383791731445, 44099.45591207383, 44920.99027777789],
  [43388.383791731445, 44099.45591207383, 44920.99027777789],
  [43388.383791731445, 44099.45591207383, 44920.99027777789],
  [43388.383791731445, 44099.45591207383, 44920.99027777789],
  [43388.383791731445, 44099.45591207383, 44920.99027777789],
  [43388.383791731445, 44099.45591207383, 44920.99027777789],
  [43388.383791731445, 44099.45591207383, 44920.99027777789],
  [43388.408632032675, 44099.45726969802, 44920.99027777789],
  [43388.346084432, 44099.45064558078, 44920.99027777789],
  [43388.35087832446, 44099.451435719064, 44920.99027777789],
  [43388.368657927655, 44099.45314695144, 44920.99027777789],
  [43388.368657927655, 44099.45314695144, 44920.99027777789],
  [43388.368657927655, 44099.45314695144, 44920.99027777789],
  [43388.368657927655, 44099.45314695144, 44920.99027777789],
  [43388.368657927655, 44099.45314695144, 44920.99027777789],
  [43388.38018698197, 44099.45411939744, 44920.98997879902],
  [43388.37352437623, 44099.453495186164, 44920.98993722775],
  [43388.37352437623, 44099.453495186164, 44920.98993722775],
  [43388.37352437623, 44099.453495186164, 44920.98993722775],
  [43388.36794380201, 44099.45314375377, 44920.990277777884],
  [43388.36794380201, 44099.45314375377, 44920.990277777884],
  [43388.37099088016, 44099.453412763745, 44920.990260873026],
  [43388.369157978275, 44099.45323003389, 44920.990271185954]],
 51)

dates_REAL

[43394.38055555569, 43937.503472222015, 44555.99027777789]

GA Result Analysis

deltaV_GA, fuel_mass_GA, data_dict_GA = find_deltaV_mission(*best_individual)
deltaV_GA, fuel_mass_GA, data_dict_GA

(9.388030185988406,
 1069.9564284063217,
 {'deltaV_injection (km/s)': 9.388030110672208,
  'deltaV_jupiter (km/s)': 7.531619772294107e-08,
  'rp_jupiter (km)': 848166.6383611013,
  'tof (days)': 1532.6211132076787})

date_departure_mjd_GA = best_individual[0]
date_flyby_jupiter_mjd_GA = best_individual[1]
date_flyby_saturn_mjd_GA = best_individual[2]

dates_list_GA = best_individual
dates_list_GA

[43388.369157978275, 44099.45323003389, 44920.990271185954]

print(f"Difference for date_departure: {date_departure_mjd - date_departure_mjd_GA}")
print(f"Difference for date_flyby_jupiter: {date_flyby_jupiter_mjd - date_flyby_jupiter_mjd_GA}")
print(f"Difference for date_flyby_saturn: {date_flyby_saturn_mjd - date_flyby_saturn_mjd_GA}\n")

change_in_dates_list_GA = [
    (date_departure_mjd - date_departure_mjd_GA),
    (date_flyby_jupiter_mjd - date_flyby_jupiter_mjd_GA),
    (date_flyby_saturn_mjd - date_flyby_saturn_mjd_GA)
]
change_in_dates_list_GA

Difference for date_departure: 6.011397577414755
Difference for date_flyby_jupiter: -161.94975781187532
Difference for date_flyby_saturn: -364.99999340806244

[6.011397577414755, -161.94975781187532, -364.99999340806244]

print("Earth-Jupiter:",date_flyby_jupiter_mjd_GA - date_departure_mjd_GA,"days")
print("Jupiter-Saturn:",date_flyby_saturn_mjd_GA - date_flyby_jupiter_mjd_GA,"days")

Earth-Jupiter: 711.0840720556153 days
Jupiter-Saturn: 821.5370411520635 days

date_departure_GA = MJD_to_TT_calander_date(date_departure_mjd_GA)
date_flyby_jupiter_GA = MJD_to_TT_calander_date(date_flyby_jupiter_mjd_GA)
date_flyby_saturn_GA = MJD_to_TT_calander_date(date_flyby_saturn_mjd_GA)

print("Date of Departure (Min):", MJD_to_TT_calander_date(date_departure_mjd_GA))

print("Date of Jupiter Flyby (Min):", MJD_to_TT_calander_date(date_flyby_jupiter_mjd_GA))

print("Date of Saturn Flyby (Min):", MJD_to_TT_calander_date(date_flyby_saturn_mjd_GA))

print([date_departure_GA, date_flyby_jupiter_GA, date_flyby_saturn_GA])

Date of Departure (Min): 1977-09-02 08:51:35.249
Date of Jupiter Flyby (Min): 1979-08-14 10:52:39.075
Date of Saturn Flyby (Min): 1981-11-12 23:45:59.430
['1977-09-02 08:51:35.249', '1979-08-14 10:52:39.075', '1981-11-12 23:45:59.430']

Objective Function vs No. of Generations

generations = list(range(len(best_solutions_per_generation)))  # X-axis (Generation numbers)
deltaV_values_GA = []  # Y-axis (Best Delta V per generation)
fuel_mass_values = []

for solution in best_solutions_per_generation:
    best_deltaV, best_fuel_mass, _ = find_deltaV_mission(*solution)  # Compute Delta V for each best solution
    deltaV_values_GA.append(best_deltaV)
    fuel_mass_values.append(best_fuel_mass)

# Plot the data
fig, ax = plt.subplots(figsize=(9, 6))

plt.plot(generations[:], deltaV_values_GA[:], marker='.', linestyle='--', label="Best $\Delta V$ [km/s] per Generation")

plt.xlabel("No. of Generations")
plt.ylabel("Objective Function")

plt.title("Objective Function vs No. of Generations")

plt.legend(loc="best")

# Improve grid and ticks
ax.grid(which='major', color='#DDDDDD', linewidth=0.8)
ax.grid(which='minor', color='#EEEEEE', linestyle=':', linewidth=0.5)
ax.minorticks_on()

# Customize tick parameters
ax.tick_params(axis='both', which='major', labelsize=12)
ax.tick_params(axis='both', which='minor', labelsize=10)

plt.tight_layout()

# Show plot
plt.show()

Fitness Function vs No. of Generations

fitness_values_deltaV = 1/np.array(deltaV_values_GA)
fitness_values_fuel_mass = 1/np.array(fuel_mass_values)

# Plot the data
fig, ax = plt.subplots(figsize=(9, 6))

plt.plot(generations, fitness_values_deltaV, marker='.', linestyle='--', label="Fitness Function $(1/\Delta V)$ per Generation")
#plt.plot(generations, fitness_values_fuel_mass, marker='.', linestyle='--', label="Fitness Function $(1/M_f)$ per Generation")

plt.xlabel("No. of Generations", fontsize=12)
plt.ylabel("Fitness Function", fontsize=12)

plt.title(r"\textbf{Fitness Function vs No. of Generations}", pad=10, fontsize=14)

#plt.legend(loc="best")

# Improve grid and ticks
ax.grid(which='major', color='#DDDDDD', linewidth=0.8)
ax.grid(which='minor', color='#EEEEEE', linestyle=':', linewidth=0.5)
ax.minorticks_on()

# Customise tick parameters
ax.tick_params(axis='both', which='major', labelsize=12)
ax.tick_params(axis='both', which='minor', labelsize=10)

plt.tight_layout()

output_figures.save_file("voyager_I_GA_fitness_function.png", dpi=500, bbox_inches="tight")

# Show plot
plt.show()

Figure saved to: Voyager 1 - Figures/voyager_I_GA_fitness_function.png

GA_list = [date_departure_mjd_GA, date_flyby_jupiter_mjd_GA, date_flyby_saturn_mjd_GA]
GA_list_calendar = [date_departure_GA, date_flyby_jupiter_GA, date_flyby_saturn_GA]
GA_list, GA_list_calendar

([43388.369157978275, 44099.45323003389, 44920.990271185954],
 ['1977-09-02 08:51:35.249',
  '1979-08-14 10:52:39.075',
  '1981-11-12 23:45:59.430'])

Contour Graph per Generation

# Extract best solutions per generation
best_solutions = np.array(best_solutions_per_generation)

departure_dates = best_solutions[:, 0] - date_departure_mjd
jupiter_flyby_dates = best_solutions[:, 1] - date_flyby_jupiter_mjd
saturn_flyby_dates = best_solutions[:, 2] - date_flyby_saturn_mjd

# Compute deltaV for colour mapping
deltaV_values_GA = np.array([find_deltaV_mission(*sol)[0] for sol in best_solutions])

fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')

# Scatter plot of best solutions per generation
scatter = ax.scatter(departure_dates, jupiter_flyby_dates, saturn_flyby_dates,
                     c=deltaV_values_GA, cmap=parula_map(True), s=50, alpha=0.5, marker='x',
                     norm=plt.Normalize(vmin=min(deltaV_values_GA), vmax=max(deltaV_values_GA)))

ax.scatter([0], 
           [0], 
           [0], 
           color='red', s=100, marker='o', edgecolors='black', linewidth=1.2, 
           zorder=3,  # Higher zorder brings it to the front
           label=f'Real Delta-V: {deltaV_REAL:.2f} km/s\nReal Fuel Mass: {fuel_mass_REAL:.2f} kg')

ax.plot([0, departure_dates[0]], 
        [0, jupiter_flyby_dates[0]], 
        [0, saturn_flyby_dates[0]], 
        linestyle='-', color='r', markersize=6, linewidth=0.5, alpha=0.8)

ax.plot(departure_dates, jupiter_flyby_dates, saturn_flyby_dates, linestyle='-', color='r', markersize=6, linewidth=0.5, alpha=0.8)

ax.scatter(date_departure_mjd_GA - date_departure_mjd, 
           date_flyby_jupiter_mjd_GA - date_flyby_jupiter_mjd, 
           date_flyby_saturn_mjd_GA - date_flyby_saturn_mjd, 
           color='fuchsia', s=150, marker='*', edgecolors='black', linewidth=1.2, 
           zorder=3,  # Higher zorder brings it to the front
           label=f'Minimum Delta-V: {deltaV_GA:.4f} km/s\nMinimum Fuel Mass: {fuel_mass_GA:.2f} kg')

# Colour bar
# cbar = fig.colorbar(scatter, ax=ax, shrink=0.6, pad=0.1)
# cbar.set_label('Total Delta-V (km/s)')

cbar = plt.colorbar(scatter, ax=ax, label='$\Delta$V (km/s)')

# Set labels with increased padding
ax.set_xlabel('Change in Earth Departure Epoch (days)')
ax.set_ylabel('Change in Jupiter Flyby Epoch (days)')
ax.set_zlabel('Change in Saturn Flyby Epoch (days)')

# Title
ax.set_title('3D Visualisation of Genetic Algorithm\'s Best Solutions per Generation ($\pm$ 365 days bounds)')

# Set axis limits
#limit = 365
#ax.set_xlim(-limit, limit)
#ax.set_ylim(-limit, limit)
#ax.set_zlim(-limit, limit)

ax.legend()
plt.tight_layout()

output_figures.save_file("voyager_I_GA_per_generation_3D.png", dpi=500, bbox_inches="tight")

# Show plot
plt.show()

Figure saved to: Voyager 1 - Figures/voyager_I_GA_per_generation_3D.png

best_solutions_per_generation = np.array(best_solutions_per_generation)
best_solutions_per_generation[:,0]

array([43395.24936125, 43395.24936125, 43395.24936125, 43387.26530285,
       43383.92821876, 43386.59838086, 43390.38322715, 43390.38322715,
       43388.18242638, 43388.1600472 , 43388.1600472 , 43388.1600472 ,
       43388.43621713, 43388.43621713, 43388.43621713, 43388.5069545 ,
       43388.5069545 , 43388.5069545 , 43388.5069545 , 43388.3350047 ,
       43388.21305094, 43388.21305094, 43388.21305094, 43388.21305094,
       43388.27910973, 43388.30055328, 43388.30055328, 43388.30055328,
       43388.38379173, 43388.38379173, 43388.38379173, 43388.38379173,
       43388.38379173, 43388.38379173, 43388.38379173, 43388.40863203,
       43388.34608443, 43388.35087832, 43388.36865793, 43388.36865793,
       43388.36865793, 43388.36865793, 43388.36865793, 43388.38018698,
       43388.37352438, 43388.37352438, 43388.37352438, 43388.3679438 ,
       43388.3679438 , 43388.37099088, 43388.36915798])

# Extract the population for the chosen generation
selected_population = population_per_generation[0]

# Extract dates and normalise relative to nominal mission dates
departure_dates_GA = np.array([ind[0] for ind in selected_population]) - date_departure_mjd
jupiter_flyby_dates_GA = np.array([ind[1] for ind in selected_population]) - date_flyby_jupiter_mjd
saturn_flyby_dates_GA = np.array([ind[2] for ind in selected_population]) - date_flyby_saturn_mjd

# Compute deltaV for coloring
deltaV_values_GA = np.array([find_deltaV_mission(*ind)[0] for ind in selected_population])

# Create 3D scatter plot
fig = plt.figure(figsize=(12, 10))
ax = fig.add_subplot(111, projection='3d')

scatter = ax.scatter(departure_dates_GA, jupiter_flyby_dates_GA, saturn_flyby_dates_GA,
                         c=deltaV_values_GA, cmap=parula_map(True), s=10, alpha=0.4,
                         norm=plt.Normalize(vmin=min(deltaV_values_GA), vmax=min(deltaV_values_GA) * 1.5))

ax.scatter(best_solutions_per_generation[:-6,0] - date_departure_mjd,
          best_solutions_per_generation[:-6,1] - date_flyby_jupiter_mjd,
          best_solutions_per_generation[:-6,2] - date_flyby_saturn_mjd,
          color='#81DB24', s=20, marker='s', edgecolors='black', linewidth=1 )

ax.scatter([0], 
           [0], 
           [0], 
           color='red', s=200, marker='o', edgecolors='black', linewidth=1.2, 
           label=f'Real Delta-V: {deltaV_REAL:.2f} km/s\nReal Fuel Mass: {fuel_mass_REAL:.2f} kg')

ax.scatter(date_departure_mjd_GA - date_departure_mjd, 
           date_flyby_jupiter_mjd_GA - date_flyby_jupiter_mjd, 
           date_flyby_saturn_mjd_GA - date_flyby_saturn_mjd, 
           color='fuchsia', s=300, marker='*', edgecolors='black', linewidth=1.2, 
           label=f'Minimum Delta-V: {deltaV_GA:.4f} km/s\nMinimum Fuel Mass: {fuel_mass_GA:.2f} kg')

ax.plot([0, departure_dates[0]], 
        [0, jupiter_flyby_dates[0]], 
        [0, saturn_flyby_dates[0]], 
        linestyle='--', color='black', markersize=6, linewidth=2, alpha=1)

ax.plot(departure_dates, jupiter_flyby_dates, saturn_flyby_dates, linestyle='--', color='black', markersize=6, linewidth=2, alpha=1)

# Set labels with increased padding
ax.set_xlabel('Change in Earth Departure Epoch (days)', labelpad=5, fontsize=12)
ax.set_ylabel('Change in Jupiter Flyby Epoch (days)', labelpad=5, fontsize=12)
ax.set_zlabel('Change in Saturn Flyby Epoch (days)', labelpad=5, fontsize=12)

# Set title with increased padding
ax.set_title(r'\textbf{3D Visualisation of Genetic Algorithm\'s Initial Population Problem Space ($\pm$ 365 days bounds)}', 
             fontsize=14) 

# Format the scientific notation to be more readable
for axis in [ax.xaxis, ax.yaxis, ax.zaxis]:
    formatter = ScalarFormatter(useOffset=False)
    formatter.set_scientific(True)
    formatter.set_powerlimits((-3, 3))  # Only use scientific notation for exponents outside this range
    axis.set_major_formatter(formatter)

# Position the offset text for better visibility
ax.xaxis.get_offset_text().set_position((0, -0.1))
ax.yaxis.get_offset_text().set_position((0, -0.1))
ax.zaxis.get_offset_text().set_position((0, -0.1))

# Ensure the exponent text is horizontally aligned
ax.xaxis.get_offset_text().set_ha('left')
ax.yaxis.get_offset_text().set_ha('left')
ax.zaxis.get_offset_text().set_ha('left')

# Add colorbar with adjusted position
cbar = plt.colorbar(scatter, ax=ax, label='$\Delta$V (km/s)')
cbar.ax.set_ylabel('$\Delta$V (km/s)', fontsize=12)

ax.legend()
plt.tight_layout()

output_figures.save_file("voyager_I_GA_inital_pop_3D.png", dpi=500, bbox_inches="tight")

plt.show()

Figure saved to: Voyager 1 - Figures/voyager_I_GA_inital_pop_3D.png

def plot_3d_ga_contour_generation(population_per_generation, generation_idx):
    # Make sure the generation index is valid
    if generation_idx < 0 or generation_idx >= len(population_per_generation):
        raise ValueError(f"Generation index must be between 0 and {len(population_per_generation)-1}")
    
    fig = plt.figure(figsize=(10, 8))
    ax = fig.add_subplot(111, projection='3d')
    
    # Get the selected population
    selected_population = population_per_generation[generation_idx]
    
    # Extract dates and normalise relative to nominal mission dates
    departure_dates_GA = np.array([ind[0] for ind in selected_population]) - date_departure_mjd
    jupiter_flyby_dates_GA = np.array([ind[1] for ind in selected_population]) - date_flyby_jupiter_mjd
    saturn_flyby_dates_GA = np.array([ind[2] for ind in selected_population]) - date_flyby_saturn_mjd
    
    deltaV_values_GA = np.array([find_deltaV_mission(*ind)[0] for ind in selected_population])
    
    # Create scatter plot
    scatter = ax.scatter(departure_dates_GA, jupiter_flyby_dates_GA, saturn_flyby_dates_GA,
                         c=deltaV_values_GA, cmap=parula_map(True), s=10, alpha=0.3,
                         norm=plt.Normalize(vmin=min(deltaV_values_GA), vmax=min(deltaV_values_GA) * 1.5))
    
    # Mark the best solution for the current generation
    best_individual = selected_population[0]  # Best individual of the generation
    ax.scatter(best_individual[0] - date_departure_mjd,
               best_individual[1] - date_flyby_jupiter_mjd,
               best_individual[2] - date_flyby_saturn_mjd,
               color='red', s=100, marker='x', label="Best Solution")
    
    # Mark the real solution
    ax.scatter([0], [0], [0],
               color='red', s=100, marker='o', edgecolors='black', linewidth=1.2,
               label=f'Real Delta-V: {deltaV_REAL:.2f} km/s\nReal Fuel Mass: {fuel_mass_REAL:.2f} kg')
    
    # Set labels with increased padding
    ax.set_xlabel('Change in Earth Departure Epoch (days)', labelpad=5, fontsize=12)
    ax.set_ylabel('Change in Jupiter Flyby Epoch (days)', labelpad=5, fontsize=12)
    ax.set_zlabel('Change in Saturn Flyby Epoch (days)', labelpad=5, fontsize=12)
    
    # Set title with increased padding
    title_text = f'\\textbf{{3D Visualisation of Genetic Algorithm Problem Space (Generation {generation_idx})}}'
    ax.set_title(r'{}'.format(title_text), fontsize=14) 
    
    # Format the scientific notation to be more readable
    for axis in [ax.xaxis, ax.yaxis, ax.zaxis]:
        formatter = ScalarFormatter(useOffset=False)
        formatter.set_scientific(True)
        formatter.set_powerlimits((-3, 3))  # Only use scientific notation for exponents outside this range
        axis.set_major_formatter(formatter)

    # Position the offset text for better visibility
    ax.xaxis.get_offset_text().set_position((0, -0.1))
    ax.yaxis.get_offset_text().set_position((0, -0.1))
    ax.zaxis.get_offset_text().set_position((0, -0.1))

    # Ensure the exponent text is horizontally aligned
    ax.xaxis.get_offset_text().set_ha('left')
    ax.yaxis.get_offset_text().set_ha('left')
    ax.zaxis.get_offset_text().set_ha('left')
    
    # Add legend
    ax.legend()
    
    # Set axis limits
    limit = 365
    ax.set_xlim(-limit, limit)
    ax.set_ylim(-limit, limit)
    ax.set_zlim(-limit, limit)
    
    # Add colorbar
    ccbar = plt.colorbar(scatter, ax=ax, label='$\Delta$V (km/s)')
    cbar.ax.set_ylabel('$\Delta$V (km/s)', fontsize=12)
    
    plt.tight_layout()
    output_figures.save_file(f"voyager_I_GA_generation_{generation_idx}.png", dpi=500, bbox_inches="tight")
    
    plt.show()
    
    return fig

fig = plot_3d_ga_contour_generation(population_per_generation, 0)
fig = plot_3d_ga_contour_generation(population_per_generation, 1)
fig = plot_3d_ga_contour_generation(population_per_generation, 5)
fig = plot_3d_ga_contour_generation(population_per_generation, 10)
fig = plot_3d_ga_contour_generation(population_per_generation, 15)
fig = plot_3d_ga_contour_generation(population_per_generation, 20)
fig = plot_3d_ga_contour_generation(population_per_generation, 25)
fig = plot_3d_ga_contour_generation(population_per_generation, 50)

Figure saved to: Voyager 1 - Figures/voyager_I_GA_generation_0.png

Figure saved to: Voyager 1 - Figures/voyager_I_GA_generation_1.png

Figure saved to: Voyager 1 - Figures/voyager_I_GA_generation_5.png

Figure saved to: Voyager 1 - Figures/voyager_I_GA_generation_10.png

Figure saved to: Voyager 1 - Figures/voyager_I_GA_generation_15.png

Figure saved to: Voyager 1 - Figures/voyager_I_GA_generation_20.png

Figure saved to: Voyager 1 - Figures/voyager_I_GA_generation_25.png

Figure saved to: Voyager 1 - Figures/voyager_I_GA_generation_50.png

deltaV_values_for_each_gen = [find_deltaV_mission(*solution)[0] for solution in best_solutions_per_generation]
deltaV_values_for_each_gen

[9.673398226512557,
 9.673398226512557,
 9.673398226512557,
 9.658887436753394,
 9.541724931289206,
 9.458554967466874,
 9.423885054018644,
 9.423885054018644,
 9.414175975790753,
 9.392525795434343,
 9.392525795434343,
 9.392525795434343,
 9.389652687163839,
 9.389652687163839,
 9.389652687163839,
 9.388551243485637,
 9.388551243485637,
 9.388551243485637,
 9.388551243485637,
 9.38833146299216,
 9.388190890444521,
 9.388190890444521,
 9.388190890444521,
 9.388190890444521,
 9.388140077046343,
 9.388059693351178,
 9.388059693351178,
 9.388059693351178,
 9.388041807870483,
 9.388041807870483,
 9.388041807870483,
 9.388041807870483,
 9.388041807870483,
 9.388041807870483,
 9.388041807870483,
 9.388038809010995,
 9.38803843842005,
 9.388033358155313,
 9.388030674653933,
 9.388030674653933,
 9.388030674653933,
 9.388030674653933,
 9.388030674653933,
 9.388030627581857,
 9.388030408862573,
 9.388030408862573,
 9.388030408862573,
 9.388030234293744,
 9.388030234293744,
 9.388030194026426,
 9.388030185988406]

GA Results --> Fmin

%%time 

# Start timing
start_time = time.time()

# Run optimisation using `fmin` with manual bounds enforcement
optimal_dates = fmin(objective_function, GA_list, disp=True)

# Extract optimised dates
date_departure_mjd_GA_FMIN_COMBINED, date_flyby_jupiter_mjd_GA_FMIN_COMBINED, date_flyby_saturn_mjd_GA_FMIN_COMBINED = optimal_dates

# Compute Delta-V and fuel mass at optimised dates
deltaV_GA_FMIN_COMBINED, fuel_mass_GA_FMIN_COMBINED, data_dict_GA_FMIN_COMBINED = find_deltaV_mission(date_departure_mjd_GA_FMIN_COMBINED, date_flyby_jupiter_mjd_GA_FMIN_COMBINED, date_flyby_saturn_mjd_GA_FMIN_COMBINED)

# Display results
print(f"Minimum Delta-V: {deltaV_GA_FMIN_COMBINED:.4f} km/s")
print(f"Optimal Departure Date (MJD): {date_departure_mjd_GA_FMIN_COMBINED:.2f}")
print(f"Optimal Jupiter Flyby Date (MJD): {date_flyby_jupiter_mjd_GA_FMIN_COMBINED:.2f}")
print(f"Optimal Saturn Flyby Date (MJD): {date_flyby_saturn_mjd_GA_FMIN_COMBINED:.2f}")

# Convert to calendar dates
print("\nIn calendar dates:")
print(f"Departure: {MJD_to_TT_calander_date(date_departure_mjd_GA_FMIN_COMBINED)}")
print(f"Jupiter flyby: {MJD_to_TT_calander_date(date_flyby_jupiter_mjd_GA_FMIN_COMBINED)}")
print(f"Saturn flyby: {MJD_to_TT_calander_date(date_flyby_saturn_mjd_GA_FMIN_COMBINED)}")

# End timing
end_time = time.time()
execution_time_GA_FMIN_COMBINED = end_time - start_time
print(f"\nExecution Time: {execution_time_GA_FMIN_COMBINED:.4f} seconds")

Optimization terminated successfully.
         Current function value: 9.388030
         Iterations: 101
         Function evaluations: 193
Minimum Delta-V: 9.3880 km/s
Optimal Departure Date (MJD): 43388.37
Optimal Jupiter Flyby Date (MJD): 44099.45
Optimal Saturn Flyby Date (MJD): 44920.99

In calendar dates:
Departure: 1977-09-02 08:51:39.410
Jupiter flyby: 1979-08-14 10:52:38.296
Saturn flyby: 1981-11-12 23:45:54.227

Execution Time: 0.5073 seconds
CPU times: user 491 ms, sys: 22.1 ms, total: 513 ms
Wall time: 507 ms

deltaV_GA_FMIN_COMBINED, fuel_mass_GA_FMIN_COMBINED, data_dict_GA_FMIN_COMBINED = find_deltaV_mission(date_departure_mjd_GA_FMIN_COMBINED, date_flyby_jupiter_mjd_GA_FMIN_COMBINED, date_flyby_saturn_mjd_GA_FMIN_COMBINED)
deltaV_GA_FMIN_COMBINED, fuel_mass_GA_FMIN_COMBINED, data_dict_GA_FMIN_COMBINED

(9.388030168438553,
 1069.9564270478534,
 {'deltaV_injection (km/s)': 9.388030134667423,
  'deltaV_jupiter (km/s)': 3.377112989255693e-08,
  'rp_jupiter (km)': 848166.5346901253,
  'tof (days)': 1532.621004831104})

change_in_dates_list_FMIN_COMBINED = [
    (date_departure_mjd - date_departure_mjd_GA_FMIN_COMBINED),
    (date_flyby_jupiter_mjd - date_flyby_jupiter_mjd_GA_FMIN_COMBINED),
    (date_flyby_saturn_mjd - date_flyby_saturn_mjd_GA_FMIN_COMBINED)
]
change_in_dates_list_FMIN_COMBINED

[6.011349423162756, -161.94974879429355, -364.9999331857398]

print("Earth-Jupiter:",date_flyby_jupiter_mjd_GA_FMIN_COMBINED - date_departure_mjd_GA_FMIN_COMBINED,"days")
print("Jupiter-Saturn:",date_flyby_saturn_mjd_GA_FMIN_COMBINED - date_flyby_jupiter_mjd_GA_FMIN_COMBINED,"days")

Earth-Jupiter: 711.0840148837815 days
Jupiter-Saturn: 821.5369899473226 days

date_departure_GA_FMIN_COMBINED = MJD_to_TT_calander_date(date_departure_mjd_GA_FMIN_COMBINED)
date_flyby_jupiter_GA_FMIN_COMBINED = MJD_to_TT_calander_date(date_flyby_jupiter_mjd_GA_FMIN_COMBINED)
date_flyby_saturn_GA_FMIN_COMBINED = MJD_to_TT_calander_date(date_flyby_saturn_mjd_GA_FMIN_COMBINED)

print("Date of Departure (Min):", MJD_to_TT_calander_date(date_departure_mjd_GA_FMIN_COMBINED))

print("Date of Jupiter Flyby (Min):", MJD_to_TT_calander_date(date_flyby_jupiter_mjd_GA_FMIN_COMBINED))

print("Date of Saturn Flyby (Min):", MJD_to_TT_calander_date(date_flyby_saturn_mjd_GA_FMIN_COMBINED))

Date of Departure (Min): 1977-09-02 08:51:39.410
Date of Jupiter Flyby (Min): 1979-08-14 10:52:38.296
Date of Saturn Flyby (Min): 1981-11-12 23:45:54.227

Graphs

Comparsion Graphs

def plot_VoyagerI(date_departure, date_flyby_jupiter, date_flyby_saturn, colour, label_suffix=""):
    
    # Convert input dates to Astropy Time in TDB scale
    date_departure = Time(date_departure, scale="tt").tdb
    date_flyby_jupiter = Time(date_flyby_jupiter, scale="tt").tdb
    date_flyby_saturn = Time(date_flyby_saturn, scale="tt").tdb

    date_arrival = date_flyby_saturn

    # Define planetary ephemerides
    earth = Ephem.from_body(Earth, time_range(date_departure, end=date_arrival, periods=500))
    ss_earth = Orbit.from_ephem(Sun, earth, date_departure)

    jupiter = Ephem.from_body(Jupiter, time_range(date_departure, end=date_arrival, periods=500))
    ss_jupiter = Orbit.from_ephem(Sun, jupiter, date_flyby_jupiter)

    saturn = Ephem.from_body(Saturn, time_range(date_departure, end=date_arrival, periods=500))
    ss_saturn = Orbit.from_ephem(Sun, saturn, date_arrival)

    # Solving for maneuver to Jupiter
    man_flyby_jupiter = Maneuver.lambert(ss_earth, ss_jupiter)
    ic1 = ss_earth.apply_maneuver(man_flyby_jupiter)
    ic1_end = ic1.propagate(date_flyby_jupiter)

    # Solving for maneuver to Saturn
    man_flyby_saturn = Maneuver.lambert(ic1_end, ss_saturn)
    ic2 = ic1_end.apply_maneuver(man_flyby_saturn)
    ic2_end = ic2.propagate(date_arrival)

    # Plot Earth's position at departure
    plotter.plot_body_orbit(
        Earth, date_departure, label=f"Earth Departure {label_suffix}", trail=True
    )

    # Plot cruise to Jupiter
    plotter.plot_maneuver(
        ss_earth, man_flyby_jupiter, label=f"Cruise to Jupiter {label_suffix}", color=colour
    )

    # Plot Jupiter's position at flyby
    plotter.plot_body_orbit(
        Jupiter, date_flyby_jupiter, label=f"Jupiter Flyby {label_suffix}", trail=True
    )

    # Plot cruise to Saturn
    plotter.plot_maneuver(
        ic1_end, man_flyby_saturn, label=f"Cruise to Saturn {label_suffix}", color=colour
    )

    # Plot Earth's position at departure
    plotter.plot_body_orbit(
        Earth, date_departure, label=f"Earth Departure {label_suffix}", trail=True
    )

    # Plot Jupiter's position at flyby
    plotter.plot_body_orbit(
        Jupiter, date_flyby_jupiter, label=f"Jupiter Flyby {label_suffix}", trail=True
    )

    # Plot Saturn's position at arrival
    plotter.plot_body_orbit(
        Saturn, date_flyby_saturn, label=f"Saturn Arrival {label_suffix}", trail=True
    )

2-Axis

colours = {
    "Real Mission": "crimson",     
    "Brute Force Algorithm": "cyan", 
    "FMIN": "magenta",                 
    "GA": "gold",                      
    "GA --> FMIN": "lime",         
}

# Distinct line styles to prevent visual merging
line_styles = {
    "Real Mission": "-",              
    "Brute Force Algorithm": "-",    
    "FMIN": "--",                     
    "GA": '--',#(0, (5, 1)),               
    "GA --> FMIN": ":",              
}

# Define different line widths for clarity
line_widths = {
    "Real Mission": 2,
    "Brute Force Algorithm": 4,
    "FMIN": 3,
    "GA": 2,
    "GA --> FMIN": 2
}

# Create figure and axis
fig = plt.figure(figsize=(12, 10))
ax = fig.add_subplot(111)
plotter = StaticOrbitPlotter(ax)

# Plot missions
plot_VoyagerI(date_departure, date_flyby_jupiter, date_flyby_saturn, colours["Real Mission"], label_suffix="(Real Mission)")
plot_VoyagerI(date_departure_BRUTE_MIN, date_flyby_jupiter_BRUTE_MIN, date_flyby_saturn_BRUTE_MIN, colours["Brute Force Algorithm"], label_suffix="(Brute Force Algorithm)")
plot_VoyagerI(date_departure_FMIN, date_flyby_jupiter_FMIN, date_flyby_saturn_FMIN, colours["FMIN"], label_suffix="(FMIN)")
plot_VoyagerI(date_departure_GA, date_flyby_jupiter_GA, date_flyby_saturn_GA, colours["GA"], label_suffix="(GA)")
plot_VoyagerI(date_departure_GA_FMIN_COMBINED, date_flyby_jupiter_GA_FMIN_COMBINED, date_flyby_saturn_GA_FMIN_COMBINED, colours["GA --> FMIN"], label_suffix="(GA --> FMIN)")

# Set fixed limits for primary axis
limit = 1.7e9
ax.set_xlim(-limit, limit)
ax.set_ylim(-limit, limit)
ax.set_aspect('equal')  # Square aspect ratio

ax.set_xlabel("x (km)", labelpad=5, fontsize=12)
ax.set_ylabel("y (km)", labelpad=5, fontsize=12)

# Conversion factor (1 AU = 149,597,870.691 km)
km_to_AU = 1 / 149597870.691

# Create second set of axes for AU scale
# These won't be linked to the primary axes in the problematic way
ax_top = fig.add_axes(ax.get_position(), frameon=False)
ax_top.xaxis.tick_top()
ax_top.yaxis.tick_right()
ax_top.set_xlim(ax.get_xlim()[0] * km_to_AU, ax.get_xlim()[1] * km_to_AU)
ax_top.set_ylim(ax.get_ylim()[0] * km_to_AU, ax.get_ylim()[1] * km_to_AU)
ax_top.set_xlabel("x (AU)", labelpad=5, fontsize=12)
ax_top.set_ylabel("y (AU)", labelpad=5, fontsize=12)
ax_top.xaxis.set_label_position('top')
ax_top.yaxis.set_label_position('right')

# # Add grid
ax.grid(which='major', color='#DDDDDD', linewidth=0.8)
ax.grid(which='minor', color='#EEEEEE', linestyle=':', linewidth=0.5)
ax.minorticks_on()
#ax_top.grid(which='major', color='#DDDDDD', linewidth=0.8)
#ax_top.grid(which='minor', color='#EEEEEE', linestyle=':', linewidth=0.5)
ax_top.minorticks_on()

# Customise tick parameters
ax.tick_params(axis='both', which='major', labelsize=10)
ax.tick_params(axis='both', which='minor', labelsize=8)
ax_top.tick_params(axis='both', which='major', labelsize=10)
ax_top.tick_params(axis='both', which='minor', labelsize=8)

# Add title
plt.title(r"\textbf{Voyager I Mission Trajectory Optimisation}", pad=10, fontsize=14) 

# Function to update AU axes when primary axes change
def update_au_axes(event=None):
    ax_top.set_xlim(ax.get_xlim()[0] * km_to_AU, ax.get_xlim()[1] * km_to_AU)
    ax_top.set_ylim(ax.get_ylim()[0] * km_to_AU, ax.get_ylim()[1] * km_to_AU)
    fig.canvas.draw_idle()

# Connect the update function to the 'xlim_changed' and 'ylim_changed' events
ax.callbacks.connect('xlim_changed', update_au_axes)
ax.callbacks.connect('ylim_changed', update_au_axes)

# # Define different line styles for each plot
# line_styles = {
#     "Real Mission": "-",
#     "Brute Force Algorithm": "-",
#     "FMIN": "-.",
#     "GA": "--",
#     "GA --> FMIN": ":"
# }
# # Define different line widths for clarity
# line_widths = {
#     "Real Mission": 2,
#     "Brute Force Algorithm": 4.5,
#     "FMIN": 3,
#     "GA": 2,
#     "GA --> FMIN": 2.5
# }

# Apply line styles
for line in ax.get_lines():
    label = line.get_label()
    for key in line_styles:
        if key in label:
            line.set_linestyle(line_styles[key])
            line.set_linewidth(line_widths[key])

# Create custom legend
handles = [
    Line2D(
        [0],
        [0],
        color=colours["Real Mission"],
        ls=line_styles["Real Mission"],
        lw=line_widths["Real Mission"],
        label="Real Mission",
    ),
    Line2D(
        [0],
        [0],
        color=colours["Brute Force Algorithm"],
        ls=line_styles["Brute Force Algorithm"],
        lw=line_widths["Brute Force Algorithm"],
        label="Brute Force Search",
    ),
    Line2D(
        [0],
        [0],
        color=colours["FMIN"],
        ls=line_styles["FMIN"],
        lw=line_widths["FMIN"],
        label="Fmin",
    ),
    Line2D(
        [0],
        [0],
        color=colours["GA"],
        ls=line_styles["GA"],
        lw=line_widths["GA"],
        label="Genetic Algorithm (GA)",
    ),
    Line2D(
        [0],
        [0],
        color=colours["GA --> FMIN"],
        ls=line_styles["GA --> FMIN"],
        lw=line_widths["GA --> FMIN"],
        label="GA + Fmin",
    ),
]


ax.plot(0, 0, 'o', markersize=18, markerfacecolor='#FFFF00', markeredgecolor='#FF8C00', markeredgewidth=1, zorder=100)

# Add legend
ax.legend(handles=handles, loc="upper left", ncol=1, frameon=True, edgecolor='black', fontsize = 11)

# Planet colors
planet_colors = {
    "Earth": "#3366FF",    # Blue
    "Jupiter": "#FF9933",  # Orange
    "Saturn": "#FFCC33"    # Yellow
}

# Use the new label positions provided
planet_label_positions = {
    "Earth": (-0.35e9, -0.2e9),
    "Jupiter": (0.8e9, 0.5e9),
    "Saturn": (0e9, 1.55e9)
}

# Add each planet marker and label
for planet, position in planet_label_positions.items():
    x, y = position
    
    # Use the absolute positions for labels instead of offsets
    label_x, label_y = planet_label_positions[planet]
    
    bbox_props = dict(boxstyle="round,pad=0.3", fc="white", ec="black", alpha=0.8)
    ax.annotate(planet, 
               xy=(x, y),  # Position of the planet
               xytext=(label_x, label_y),  # Absolute position for the label
               textcoords='data',  # Changed from 'offset points' to 'data'
               bbox=bbox_props,
               fontsize=12,
               fontweight='bold',
               zorder=101,  # Ensure labels appear on top of everything
               )  # Add arrow connecting label to planet

# Save figure with padding
output_figures.save_file("voyager_I_trajectory_km_AU.png", dpi=500, bbox_inches="tight")

# Show plot
plt.show()

Figure saved to: Voyager 1 - Figures/voyager_I_trajectory_km_AU.png

dates_REAL

[43394.38055555569, 43937.503472222015, 44555.99027777789]

2-Axis Zoomed

# Create figure and axis
fig = plt.figure(figsize=(12, 10))
ax = fig.add_subplot(111)
plotter = StaticOrbitPlotter(ax)

# Plot missions
plot_VoyagerI(date_departure, date_flyby_jupiter, date_flyby_saturn, colours["Real Mission"], label_suffix="(Real Mission)")
plot_VoyagerI(date_departure_BRUTE_MIN, date_flyby_jupiter_BRUTE_MIN, date_flyby_saturn_BRUTE_MIN, colours["Brute Force Algorithm"], label_suffix="(Brute Force Algorithm)")
plot_VoyagerI(date_departure_FMIN, date_flyby_jupiter_FMIN, date_flyby_saturn_FMIN, colours["FMIN"], label_suffix="(FMIN)")
plot_VoyagerI(date_departure_GA, date_flyby_jupiter_GA, date_flyby_saturn_GA, colours["GA"], label_suffix="(GA)")
plot_VoyagerI(date_departure_GA_FMIN_COMBINED, date_flyby_jupiter_GA_FMIN_COMBINED, date_flyby_saturn_GA_FMIN_COMBINED, colours["GA --> FMIN"], label_suffix="(GA --> FMIN)")

# Set fixed limits for primary axis
ax.set_xlim(-.5e9, 1e9)
ax.set_ylim(-.5e9, 1.75e9)
ax.set_aspect('equal')  # Square aspect ratio

ax.set_xlabel("x (km)", labelpad=5, fontsize=12)
ax.set_ylabel("y (km)", labelpad=5, fontsize=12)

# Conversion factor (1 AU = 149,597,870.691 km)
km_to_AU = 1 / 149597870.691

# Create second set of axes for AU scale
ax_top = fig.add_axes(ax.get_position(), frameon=False)
ax_top.xaxis.tick_top()
ax_top.yaxis.tick_right()
ax_top.set_xlim(ax.get_xlim()[0] * km_to_AU, ax.get_xlim()[1] * km_to_AU)
ax_top.set_ylim(ax.get_ylim()[0] * km_to_AU, ax.get_ylim()[1] * km_to_AU)
ax_top.set_xlabel("x (AU)", fontsize=12)
ax_top.set_ylabel("y (AU)", fontsize=12)
ax_top.xaxis.set_label_position('top')
ax_top.yaxis.set_label_position('right')

# Add grid
ax.grid(which='major', color='#DDDDDD', linewidth=0.8)
ax.grid(which='minor', color='#EEEEEE', linestyle=':', linewidth=0.5)
ax.minorticks_on()
#ax_top.grid(which='major', color='#DDDDDD', linewidth=0.8)
#ax_top.grid(which='minor', color='#EEEEEE', linestyle=':', linewidth=0.5)
ax_top.minorticks_on()

# Customize tick parameters
ax.tick_params(axis='both', which='major', labelsize=10)
ax.tick_params(axis='both', which='minor', labelsize=8)
ax_top.tick_params(axis='both', which='major', labelsize=10)
ax_top.tick_params(axis='both', which='minor', labelsize=8)

# Add title
plt.title(r"\textbf{Voyager I Mission Trajectory Optimisation}", pad=10)

# Function to update AU axes when primary axes change
def update_au_axes(event=None):
    ax_top.set_xlim(ax.get_xlim()[0] * km_to_AU, ax.get_xlim()[1] * km_to_AU)
    ax_top.set_ylim(ax.get_ylim()[0] * km_to_AU, ax.get_ylim()[1] * km_to_AU)
    fig.canvas.draw_idle()

# Connect the update function to the 'xlim_changed' and 'ylim_changed' events
ax.callbacks.connect('xlim_changed', update_au_axes)
ax.callbacks.connect('ylim_changed', update_au_axes)

# Apply line styles
for line in ax.get_lines():
    label = line.get_label()
    for key in line_styles:
        if key in label:
            line.set_linestyle(line_styles[key])
            line.set_linewidth(line_widths[key])

# Create custom legend
handles = [
    Line2D(
        [0],
        [0],
        color=colours["Real Mission"],
        ls=line_styles["Real Mission"],
        lw=line_widths["Real Mission"],
        label="Real Mission",
    ),
    Line2D(
        [0],
        [0],
        color=colours["Brute Force Algorithm"],
        ls=line_styles["Brute Force Algorithm"],
        lw=line_widths["Brute Force Algorithm"],
        label="Brute Force Search",
    ),
    Line2D(
        [0],
        [0],
        color=colours["FMIN"],
        ls=line_styles["FMIN"],
        lw=line_widths["FMIN"],
        label="Fmin",
    ),
    Line2D(
        [0],
        [0],
        color=colours["GA"],
        ls=line_styles["GA"],
        lw=line_widths["GA"],
        label="Genetic Algorithm (GA)",
    ),
    Line2D(
        [0],
        [0],
        color=colours["GA --> FMIN"],
        ls=line_styles["GA --> FMIN"],
        lw=line_widths["GA --> FMIN"],
        label="GA + Fmin",
    ),
]


ax.plot(0, 0, 'o', markersize=18, markerfacecolor='#FFFF00', markeredgecolor='#FF8C00', markeredgewidth=1, zorder=100)

# Add legend
ax.legend(handles=handles, loc="best", ncol=1, frameon=True, edgecolor='black')

# Add each planet marker and label
for planet, position in planet_label_positions.items():
    x, y = position
    
    # Use the absolute positions for labels instead of offsets
    label_x, label_y = planet_label_positions[planet]
    
    bbox_props = dict(boxstyle="round,pad=0.3", fc="white", ec="black", alpha=0.8)
    ax.annotate(planet, 
               xy=(x, y),  # Position of the planet
               xytext=(label_x, label_y),  # Absolute position for the label
               textcoords='data',  # Changed from 'offset points' to 'data'
               bbox=bbox_props,
               fontsize=12,
               fontweight='bold',
               zorder=101,  # Ensure labels appear on top of everything
               )  # Add arrow connecting label to planet

output_figures.save_file("voyager_I_trajectory_zoom.png", dpi=500, bbox_inches="tight")

# Show plot
plt.show()

Figure saved to: Voyager 1 - Figures/voyager_I_trajectory_zoom.png

Genetic Algorithm (GA) Animation

Orbital Trajectory per Generation

# Ensure real mission is first in the best solutions list
best_solutions_per_generation,len(best_solutions_per_generation)

(array([[43395.24936125, 44092.91517706, 44880.85431079],
        [43395.24936125, 44092.91517706, 44880.85431079],
        [43395.24936125, 44092.91517706, 44880.85431079],
        [43387.26530285, 44031.50633425, 44750.20799249],
        [43383.92821876, 44093.27800067, 44889.28658652],
        [43386.59838086, 44086.44605021, 44881.14774807],
        [43390.38322715, 44095.41255988, 44913.24275187],
        [43390.38322715, 44095.41255988, 44913.24275187],
        [43388.18242638, 44087.5432199 , 44892.05273861],
        [43388.1600472 , 44099.95270596, 44920.99027778],
        [43388.1600472 , 44099.95270596, 44920.99027778],
        [43388.1600472 , 44099.95270596, 44920.99027778],
        [43388.43621713, 44099.65258156, 44920.99027778],
        [43388.43621713, 44099.65258156, 44920.99027778],
        [43388.43621713, 44099.65258156, 44920.99027778],
        [43388.5069545 , 44099.42822319, 44920.99027778],
        [43388.5069545 , 44099.42822319, 44920.99027778],
        [43388.5069545 , 44099.42822319, 44920.99027778],
        [43388.5069545 , 44099.42822319, 44920.99027778],
        [43388.3350047 , 44099.36300614, 44920.80595897],
        [43388.21305094, 44099.40379716, 44920.90103294],
        [43388.21305094, 44099.40379716, 44920.90103294],
        [43388.21305094, 44099.40379716, 44920.90103294],
        [43388.21305094, 44099.40379716, 44920.90103294],
        [43388.27910973, 44099.43854316, 44920.99027778],
        [43388.30055328, 44099.44835895, 44920.99027778],
        [43388.30055328, 44099.44835895, 44920.99027778],
        [43388.30055328, 44099.44835895, 44920.99027778],
        [43388.38379173, 44099.45591207, 44920.99027778],
        [43388.38379173, 44099.45591207, 44920.99027778],
        [43388.38379173, 44099.45591207, 44920.99027778],
        [43388.38379173, 44099.45591207, 44920.99027778],
        [43388.38379173, 44099.45591207, 44920.99027778],
        [43388.38379173, 44099.45591207, 44920.99027778],
        [43388.38379173, 44099.45591207, 44920.99027778],
        [43388.40863203, 44099.4572697 , 44920.99027778],
        [43388.34608443, 44099.45064558, 44920.99027778],
        [43388.35087832, 44099.45143572, 44920.99027778],
        [43388.36865793, 44099.45314695, 44920.99027778],
        [43388.36865793, 44099.45314695, 44920.99027778],
        [43388.36865793, 44099.45314695, 44920.99027778],
        [43388.36865793, 44099.45314695, 44920.99027778],
        [43388.36865793, 44099.45314695, 44920.99027778],
        [43388.38018698, 44099.4541194 , 44920.9899788 ],
        [43388.37352438, 44099.45349519, 44920.98993723],
        [43388.37352438, 44099.45349519, 44920.98993723],
        [43388.37352438, 44099.45349519, 44920.98993723],
        [43388.3679438 , 44099.45314375, 44920.99027778],
        [43388.3679438 , 44099.45314375, 44920.99027778],
        [43388.37099088, 44099.45341276, 44920.99026087],
        [43388.36915798, 44099.45323003, 44920.99027119]]),
 51)

fig, ax = plt.subplots(figsize=(9, 7))
plt.close(fig)  # This prevents the initial empty figure from showing

plotter = StaticOrbitPlotter(ax)

# Store previously plotted trajectories
plotted_trajectories = []

# Select a colour map (viridis, plasma, coolwarm, rainbow, etc.)
cmap = plt.cm.rainbow  # Change this for different color gradients

# Flag to track if we've plotted the real mission trajectory
real_mission_plotted = False

# Function to update the plot at each generation
def update(frame):
    global real_mission_plotted
    
    # Plot the real mission trajectory only once at the first frame
    if frame == 0:
        # Plot real mission in red
        real_traj = plot_VoyagerI(date_departure, date_flyby_jupiter, date_flyby_saturn, 
                                 "red", label_suffix="(Real Mission)")
        plotted_trajectories.append(real_traj)
        real_mission_plotted = True
    else:
        # Get best trajectory for this generation
        best_traj = best_solutions_per_generation[frame-1]
        date_departure_gen, date_flyby_jupiter_gen, date_flyby_saturn_gen = best_traj
        
        # Convert MJD to calendar date
        date_departure_gen = MJD_to_TT_calander_date(date_departure_gen)
        date_flyby_jupiter_gen = MJD_to_TT_calander_date(date_flyby_jupiter_gen)
        date_flyby_saturn_gen = MJD_to_TT_calander_date(date_flyby_saturn_gen)
        
        # Normalise color based on generation number
        norm_color = cmap(frame / len(best_solutions_per_generation))
        
        # Plot new trajectory with gradient color
        traj_plot = plot_VoyagerI(date_departure_gen, date_flyby_jupiter_gen, 
                                 date_flyby_saturn_gen, norm_color, 
                                 label_suffix=f"(Gen {frame-1})")
        
        # Store the trajectory plot for reference
        plotted_trajectories.append(traj_plot)
    
    # Hide the legend during animation to prevent clutter
    ax.legend().set_visible(False)

    if frame == 0:
        ax.set_title(f"Voyager I GA Optimisation - Real Mission")
    else:
        # Set title dynamically
        ax.set_title(f"Voyager I GA Optimisation - Generation {frame-1}")
    
    return plotted_trajectories

# Improve grid and ticks
ax.grid(which='major', color='#DDDDDD', linewidth=0.8)
ax.grid(which='minor', color='#EEEEEE', linestyle=':', linewidth=0.5)
ax.minorticks_on()

# Customise tick parameters
ax.tick_params(axis='both', which='major', labelsize=12)
ax.tick_params(axis='both', which='minor', labelsize=10)

plt.tight_layout()

# Create the animation
ani = animation.FuncAnimation(fig, update, frames=len(best_solutions_per_generation)+1, 
                              repeat=False, blit=False)

# Save the animation as a GIF or MP4
#ani.save("voyager_I_GA_optimisation.mp4", writer="ffmpeg", fps=5)#, dpi=200, bitrate=-1)
#ani.save("voyager_I_GA_optimisation.gif", writer="pillow", fps=10, dpi=200)  # Save as GIF

<Figure size 640x480 with 0 Axes>

# Convert animation to HTML
html_video = ani.to_jshtml()

# Display animation inline
display(HTML(html_video))

Once Loop Reflect

fig, ax = plt.subplots(figsize=(9, 7))
plt.close(fig)  # This prevents the initial empty figure from showing

plotter = StaticOrbitPlotter(ax)

# Store previously plotted trajectories
plotted_trajectories = []

# Select a colour map (viridis, plasma, coolwarm, rainbow, etc.)
cmap = plt.cm.rainbow  # Change this for different color gradients

# Flag to track if we've plotted the real mission trajectory
real_mission_plotted = False

# Improve grid and ticks
ax.grid(which='major', color='#DDDDDD', linewidth=0.8)
ax.grid(which='minor', color='#EEEEEE', linestyle=':', linewidth=0.5)
ax.minorticks_on()

# Customise tick parameters
ax.tick_params(axis='both', which='major', labelsize=12)
ax.tick_params(axis='both', which='minor', labelsize=10)

plt.tight_layout()

# Create the animation
ani = animation.FuncAnimation(fig, update, frames=len(best_solutions_per_generation)+1, 
                              repeat=False, blit=False)

filename_GA_trajectory_per_gen = output_ani.get_path("voyager_I_GA_trajectory_per_gen.gif")

ani.save(filename_GA_trajectory_per_gen, writer="pillow", fps=10, dpi=200)

<Figure size 640x480 with 0 Axes>

def display_saved_gif(filename):
    with open(filename, 'rb') as file:
        gif_data = file.read()
    encoded = b64encode(gif_data).decode('ascii')
    html = f'<img src="data:image/gif;base64,{encoded}">'
    return HTML(html)

display(display_saved_gif(filename_GA_trajectory_per_gen))

GA Problem Space per Generation

# Function to animate the evolution of the genetic algorithm
def animate_3d_ga_contour(population_per_generation):
    fig = plt.figure(figsize=(10, 8))
    plt.close(fig)  # This prevents the initial empty figure from showing
    ax = fig.add_subplot(111, projection='3d')
    cmap = plt.cm.viridis_r  # Color map for deltaV values
    cbar = None  # Initialise colorbar variable

    def update(frame):
        nonlocal cbar
        ax.clear()
        generation_idx = frame
        selected_population = population_per_generation[generation_idx]

        # Extract dates and normalise relative to nominal mission dates
        departure_dates_GA = np.array([ind[0] for ind in selected_population]) - date_departure_mjd
        jupiter_flyby_dates_GA = np.array([ind[1] for ind in selected_population]) - date_flyby_jupiter_mjd
        saturn_flyby_dates_GA = np.array([ind[2] for ind in selected_population]) - date_flyby_saturn_mjd
        
        # Compute deltaV for coloring
        deltaV_values_GA = np.array([find_deltaV_mission(*ind)[0] for ind in selected_population])
        
        scatter = ax.scatter(departure_dates_GA, jupiter_flyby_dates_GA, saturn_flyby_dates_GA,
                             c=deltaV_values_GA, cmap=parula_map(True), s=10, alpha=0.3,
                             norm=plt.Normalize(vmin=min(deltaV_values_GA), vmax=min(deltaV_values_GA) * 1.5))

        # Mark the best solution for the current generation
        best_individual = selected_population[0]  # Best individual of the generation
        ax.scatter(best_individual[0] - date_departure_mjd,
                   best_individual[1] - date_flyby_jupiter_mjd,
                   best_individual[2] - date_flyby_saturn_mjd,
                   color='red', s=100, marker='x', label="Best Solution")

        ax.scatter([0],
                   [0],
                   [0],
                   color='red', s=100, marker='o', edgecolors='black', linewidth=1.2,
                   label=f'Real Delta-V: {deltaV_REAL:.2f} km/s\nReal Fuel Mass: {fuel_mass_REAL:.2f} kg')

        # Set labels with increased padding
        ax.set_xlabel('Change in Earth Departure Epoch (days)', labelpad=5, fontsize=11)
        ax.set_ylabel('Change in Jupiter Flyby Epoch (days)', labelpad=5, fontsize=11)
        ax.set_zlabel('Change in Saturn Flyby Epoch (days)', labelpad=5, fontsize=11)
        
        # Set title with increased padding
        title_text = f'\\textbf{{3D Visualisation of GA Problem Space (Generation {generation_idx})}}'
        ax.set_title(r'{}'.format(title_text), fontsize=14) 
        
        # Format the scientific notation to be more readable
        for axis in [ax.xaxis, ax.yaxis, ax.zaxis]:
            formatter = ScalarFormatter(useOffset=False)
            formatter.set_scientific(True)
            formatter.set_powerlimits((-3, 3))  # Only use scientific notation for exponents outside this range
            axis.set_major_formatter(formatter)
    
        # Position the offset text for better visibility
        ax.xaxis.get_offset_text().set_position((0, -0.1))
        ax.yaxis.get_offset_text().set_position((0, -0.1))
        ax.zaxis.get_offset_text().set_position((0, -0.1))
    
        # Ensure the exponent text is horizontally aligned
        ax.xaxis.get_offset_text().set_ha('left')
        ax.yaxis.get_offset_text().set_ha('left')
        ax.zaxis.get_offset_text().set_ha('left')

        ax.legend()

        limit = 365
        ax.set_xlim(-limit, limit)
        ax.set_ylim(-limit, limit)
        ax.set_zlim(-limit, limit)
        
        # Add colorbar only once
        if cbar is None:
            cbar = plt.colorbar(scatter, ax=ax, label='$\Delta$V (km/s)', pad=0.1)  # Increase padding between plot and colorbar
            cbar.set_label('$\Delta$V (km/s)', fontsize=12, labelpad=10)
    
    ani = animation.FuncAnimation(fig, update, frames=len(population_per_generation), repeat=False)
    plt.show()
    return ani

# Example Usage
ani = animate_3d_ga_contour(population_per_generation)

# Convert animation to HTML
html_video = ani.to_jshtml()

# Display animation inline
display(HTML(html_video))

/var/folders/ts/2k06pkqj7sqgrljsl3sdbxc80000gn/T/ipykernel_55087/3993243194.py:75: UserWarning: Adding colorbar to a different Figure <Figure size 1000x800 with 2 Axes> than <Figure size 640x480 with 0 Axes> which fig.colorbar is called on.
  cbar = plt.colorbar(scatter, ax=ax, label='$\Delta$V (km/s)', pad=0.1)  # Increase padding between plot and colorbar

Once Loop Reflect

<Figure size 640x480 with 0 Axes>

filename_GA_PS_per_gen = output_ani.get_path("voyager_I_GA_problem_space_per_gen.gif")

ani.save(filename_GA_PS_per_gen, writer="pillow", fps=10, dpi=200)

def display_saved_gif(filename):
    with open(filename, 'rb') as file:
        gif_data = file.read()
    encoded = b64encode(gif_data).decode('ascii')
    html = f'<img src="data:image/gif;base64,{encoded}">'
    return HTML(html)

display(display_saved_gif(filename_GA_PS_per_gen))

Final DeltaV & Fuel Mass

deltaV_REAL, fuel_mass_REAL, data_dict_REAL

(10.31241583834745,
 1139.0,
 {'deltaV_injection (km/s)': 10.303632164583048,
  'deltaV_jupiter (km/s)': 0.008783673764401811,
  'rp_jupiter (km)': 339964.07869084046,
  'tof (days)': 1161.6097222222015})

deltaV_BRUTE_MIN, fuel_mass_BRUTE_MIN, data_dict_BRUTE_MIN

(9.411191461145675,
 1071.7476431066705,
 {'deltaV_injection (km/s)': 9.40484890472681,
  'deltaV_jupiter (km/s)': 0.006342556418864831,
  'rp_jupiter (km)': 820160.096156483,
  'tof (days)': 1519.1607426303601})

deltaV_FMIN, fuel_mass_FMIN, data_dict_FMIN

(9.388030073904538,
 1069.9564197303307,
 {'deltaV_injection (km/s)': 9.388030073866002,
  'deltaV_jupiter (km/s)': 3.8536285273949034e-11,
  'rp_jupiter (km)': 848168.1721327668,
  'tof (days)': 1532.6188724847307})

deltaV_GA, fuel_mass_GA, data_dict_GA

(9.388030185988406,
 1069.9564284063217,
 {'deltaV_injection (km/s)': 9.388030110672208,
  'deltaV_jupiter (km/s)': 7.531619772294107e-08,
  'rp_jupiter (km)': 848166.6383611013,
  'tof (days)': 1532.6211132076787})

deltaV_GA_FMIN_COMBINED, fuel_mass_GA_FMIN_COMBINED, data_dict_GA_FMIN_COMBINED

(9.388030168438553,
 1069.9564270478534,
 {'deltaV_injection (km/s)': 9.388030134667423,
  'deltaV_jupiter (km/s)': 3.377112989255693e-08,
  'rp_jupiter (km)': 848166.5346901253,
  'tof (days)': 1532.621004831104})

deltaVs = {
    "REAL": deltaV_REAL,
    "Brute Force": deltaV_BRUTE_MIN,
    "Fmin": deltaV_FMIN,
    "Genetic Algorithm": deltaV_GA,
    "GA + Fmin": deltaV_GA_FMIN_COMBINED
}

for method in deltaVs:
    print(f"{method}: {deltaVs[method]:.4f} km/s")

REAL: 10.3124 km/s
Brute Force: 9.4112 km/s
Fmin: 9.3880 km/s
Genetic Algorithm: 9.3880 km/s
GA + Fmin: 9.3880 km/s

data_dict_REAL

{'deltaV_injection (km/s)': 10.303632164583048,
 'deltaV_jupiter (km/s)': 0.008783673764401811,
 'rp_jupiter (km)': 339964.07869084046,
 'tof (days)': 1161.6097222222015}

deltaV_injection_list = {
    "REAL": data_dict_REAL['deltaV_injection (km/s)'],
    "Brute Force": data_dict_BRUTE_MIN['deltaV_injection (km/s)'],
    "Fmin": data_dict_FMIN['deltaV_injection (km/s)'],
    "Genetic Algorithm": data_dict_GA['deltaV_injection (km/s)'],
    "GA + Fmin": data_dict_GA_FMIN_COMBINED['deltaV_injection (km/s)']
}

for method in deltaV_injection_list:
    print(f"{method}: {deltaV_injection_list[method]:.4f} km/s")

REAL: 10.3036 km/s
Brute Force: 9.4048 km/s
Fmin: 9.3880 km/s
Genetic Algorithm: 9.3880 km/s
GA + Fmin: 9.3880 km/s

deltaV_jupiter_flyby_list = {
    "REAL": data_dict_REAL['deltaV_jupiter (km/s)'],
    "Brute Force": data_dict_BRUTE_MIN['deltaV_jupiter (km/s)'],
    "Fmin": data_dict_FMIN['deltaV_jupiter (km/s)'],
    "Genetic Algorithm": data_dict_GA['deltaV_jupiter (km/s)'],
    "GA + Fmin": data_dict_GA_FMIN_COMBINED['deltaV_jupiter (km/s)']
}

for method in deltaV_jupiter_flyby_list:
    print(f"{method}: {deltaV_jupiter_flyby_list[method]} km/s")

REAL: 0.008783673764401811 km/s
Brute Force: 0.006342556418864831 km/s
Fmin: 3.8536285273949034e-11 km/s
Genetic Algorithm: 7.531619772294107e-08 km/s
GA + Fmin: 3.377112989255693e-08 km/s

tof_list = {
    "REAL": data_dict_REAL['tof (days)'],
    "Brute Force": data_dict_BRUTE_MIN['tof (days)'],
    "Fmin": data_dict_FMIN['tof (days)'],
    "Genetic Algorithm": data_dict_GA['tof (days)'],
    "GA + Fmin": data_dict_GA_FMIN_COMBINED['tof (days)']
}

for method in tof_list:
    print(f"{method}: {tof_list[method]:.2f} days")

REAL: 1161.61 days
Brute Force: 1519.16 days
Fmin: 1532.62 days
Genetic Algorithm: 1532.62 days
GA + Fmin: 1532.62 days

fuel_mass_values = {
    "REAL": fuel_mass_REAL,
    "Brute Force": fuel_mass_BRUTE_MIN,
    "Fmin": fuel_mass_FMIN,
    "Genetic Algorithm": fuel_mass_GA,
    "GA + Fmin": fuel_mass_GA_FMIN_COMBINED
}

for method in fuel_mass_values:
    print(f"{method}: {fuel_mass_values[method]:.2f} kg")

REAL: 1139.00 kg
Brute Force: 1071.75 kg
Fmin: 1069.96 kg
Genetic Algorithm: 1069.96 kg
GA + Fmin: 1069.96 kg

execution_time_dict = {
    "Brute Force": execution_time_BRUTE_FORCE,
    "Fmin": execution_time_FMIN,
    "Genetic Algorithm": execution_time_GA,
    "GA + Fmin": execution_time_GA_FMIN_COMBINED
}

for method, time in execution_time_dict.items():
    print(f"{method}: {time:.2f} seconds")

Brute Force: 412.90 seconds
Fmin: 1.14 seconds
Genetic Algorithm: 363.45 seconds
GA + Fmin: 0.51 seconds

from datetime import datetime

def format_dates_to_standard(date_list):
    formatted_dates = {}
    
    for method in date_list:
        # Check if the date is already a datetime object
        if isinstance(date_list[method], datetime):
            formatted_date = date_list[method].strftime("%d/%m/%Y %H:%M:%S")
        else:
            try:
                # Try to parse the string into a datetime object
                format_string = "%Y-%m-%d %H:%M:%S.%f" if "." in str(date_list[method]) else "%Y-%m-%d %H:%M"
                dt = datetime.strptime(str(date_list[method]), format_string)
                formatted_date = dt.strftime("%d/%m/%Y %H:%M:%S")
            except ValueError:
                # If parsing fails, keep the original value
                formatted_date = str(date_list[method])
                print(f"Warning: Could not parse date for {method}: {date_list[method]}")
        
        formatted_dates[method] = formatted_date
    
    return formatted_dates

date_departure_list = {
    "REAL": date_departure,
    "Brute Force": date_departure_BRUTE_MIN,
    "Fmin": date_departure_FMIN,
    "Genetic Algorithm": date_departure_GA,
    "GA + Fmin": date_departure_GA_FMIN_COMBINED
}

for method in date_departure_list:
    print(f"{method}: {date_departure_list[method]}")

print()

format_dates_to_standard(date_departure_list)

REAL: 1977-09-08 09:08
Brute Force: 1977-08-31 22:21:28.163
Fmin: 1977-09-02 08:54:49.416
Genetic Algorithm: 1977-09-02 08:51:35.249
GA + Fmin: 1977-09-02 08:51:39.410

{'REAL': '08/09/1977 09:08:00',
 'Brute Force': '31/08/1977 22:21:28',
 'Fmin': '02/09/1977 08:54:49',
 'Genetic Algorithm': '02/09/1977 08:51:35',
 'GA + Fmin': '02/09/1977 08:51:39'}

from datetime import datetime

date_flyby_jupiter_list = {
    "REAL": date_flyby_jupiter,
    "Brute Force": date_flyby_jupiter_BRUTE_MIN,
    "Fmin": date_flyby_jupiter_FMIN,
    "Genetic Algorithm": date_flyby_jupiter_GA,
    "GA + Fmin": date_flyby_jupiter_GA_FMIN_COMBINED
}

for method in date_flyby_jupiter_list:
    print(f"{method}: {date_flyby_jupiter_list[method]}")

print()

format_dates_to_standard(date_flyby_jupiter_list)

REAL: 1979-03-05 12:05
Brute Force: 1979-08-08 22:22:08.571
Fmin: 1979-08-14 10:52:57.591
Genetic Algorithm: 1979-08-14 10:52:39.075
GA + Fmin: 1979-08-14 10:52:38.296

{'REAL': '05/03/1979 12:05:00',
 'Brute Force': '08/08/1979 22:22:08',
 'Fmin': '14/08/1979 10:52:57',
 'Genetic Algorithm': '14/08/1979 10:52:39',
 'GA + Fmin': '14/08/1979 10:52:38'}

from datetime import datetime

date_flyby_saturn_list = {
    "REAL": date_flyby_saturn,
    "Brute Force": date_flyby_saturn_BRUTE_MIN,
    "Fmin": date_flyby_saturn_FMIN,
    "Genetic Algorithm": date_flyby_saturn_GA,
    "GA + Fmin": date_flyby_saturn_GA_FMIN_COMBINED
}

for method in date_flyby_saturn_list:
    print(f"{method}: {date_flyby_saturn_list[method]}")

print()

format_dates_to_standard(date_flyby_saturn_list)

REAL: 1980-11-12 23:46
Brute Force: 1981-10-29 02:12:56.327
Fmin: 1981-11-12 23:45:59.999
Genetic Algorithm: 1981-11-12 23:45:59.430
GA + Fmin: 1981-11-12 23:45:54.227

{'REAL': '12/11/1980 23:46:00',
 'Brute Force': '29/10/1981 02:12:56',
 'Fmin': '12/11/1981 23:45:59',
 'Genetic Algorithm': '12/11/1981 23:45:59',
 'GA + Fmin': '12/11/1981 23:45:54'}

Comparsion Table

# Data
methods = ["REAL", "Brute Force", "Genetic Algorithm", "Fmin", "GA + Fmin"]

data = {
    "Total Delta-V (km/s)": [deltaVs[method] for method in methods],
    "Injection Delta-V (km/s)": [deltaV_injection_list[method] for method in methods],
    "Jupiter Flyby Delta-V (km/s)": [deltaV_jupiter_flyby_list[method] for method in methods],
    "Total Fuel Mass (kg)": [fuel_mass_values[method] for method in methods],
    "TOF (days)": [tof_list[method] for method in methods],
    "Departure Change (days)": np.array([0, change_in_dates_list_BRUTE_MIN[0], change_in_dates_list_FMIN[0], change_in_dates_list_GA[0], change_in_dates_list_FMIN_COMBINED[0]])*(-1),
    "Jupiter Flyby Change (days)": np.array([0, change_in_dates_list_BRUTE_MIN[1], change_in_dates_list_FMIN[1], change_in_dates_list_GA[1], change_in_dates_list_FMIN_COMBINED[1]])*(-1),
    "Saturn Flyby Change (days)": np.array([0, change_in_dates_list_BRUTE_MIN[2], change_in_dates_list_FMIN[2], change_in_dates_list_GA[2], change_in_dates_list_FMIN_COMBINED[2]])*(-1),
    "Execution Time (s)": [execution_time_dict.get(method, None) for method in methods]
}

df = pd.DataFrame(data, index=methods)
df

	Total Delta-V (km/s)	Injection Delta-V (km/s)	Jupiter Flyby Delta-V (km/s)	Total Fuel Mass (kg)	TOF (days)	Departure Change (days)	Jupiter Flyby Change (days)	Saturn Flyby Change (days)	Execution Time (s)
REAL	10.312416	10.303632	8.783674e-03	1139.000000	1161.609722	-0.000000	-0.000000	-0.000000	NaN
Brute Force	9.411191	9.404849	6.342556e-03	1071.747643	1519.160743	-7.448980	156.428571	350.102041	412.901764
Genetic Algorithm	9.388030	9.388030	7.531620e-08	1069.956428	1532.621113	-6.009150	161.949972	365.000000	363.453997
Fmin	9.388030	9.388030	3.853629e-11	1069.956420	1532.618872	-6.011398	161.949758	364.999993	1.144969
GA + Fmin	9.388030	9.388030	3.377113e-08	1069.956427	1532.621005	-6.011349	161.949749	364.999933	0.507342

df = pd.DataFrame(data, index=methods)
df_transposed = df.transpose()
#df_transposed.to_csv('voyager_mission_comparison.csv')
df_transposed

	REAL	Brute Force	Genetic Algorithm	Fmin	GA + Fmin
Total Delta-V (km/s)	10.312416	9.411191	9.388030e+00	9.388030e+00	9.388030e+00
Injection Delta-V (km/s)	10.303632	9.404849	9.388030e+00	9.388030e+00	9.388030e+00
Jupiter Flyby Delta-V (km/s)	0.008784	0.006343	7.531620e-08	3.853629e-11	3.377113e-08
Total Fuel Mass (kg)	1139.000000	1071.747643	1.069956e+03	1.069956e+03	1.069956e+03
TOF (days)	1161.609722	1519.160743	1.532621e+03	1.532619e+03	1.532621e+03
Departure Change (days)	-0.000000	-7.448980	-6.009150e+00	-6.011398e+00	-6.011349e+00
Jupiter Flyby Change (days)	-0.000000	156.428571	1.619500e+02	1.619498e+02	1.619497e+02
Saturn Flyby Change (days)	-0.000000	350.102041	3.650000e+02	3.650000e+02	3.649999e+02
Execution Time (s)	NaN	412.901764	3.634540e+02	1.144969e+00	5.073421e-01

deltaV_values_method = [deltaVs[name] for name in methods]
execution_methods = [m for m in methods if m in execution_time_dict]
execution_times = [execution_time_dict[name] for name in execution_methods]

# Convert to numpy arrays for plotting
deltaV_values_method = np.array(deltaV_values_method, dtype=float)
execution_times = np.array(execution_times, dtype=float)

fig, ax1 = plt.subplots(figsize=(8, 6))

# Plot delta-V
ax1.set_xlabel("Optimisation Method")
ax1.set_ylabel("$\Delta V$ (km/s)", color='tab:blue')
ax1.plot(methods, deltaV_values_method, marker='o', linestyle='--', color='tab:blue', label="$\Delta V$")
ax1.tick_params(axis='y', labelcolor='tab:blue')
ax1.set_xticks(range(len(methods)))
ax1.set_xticklabels(methods, ha='center', rotation=20, fontsize=12)

# Secondary axis for execution time
ax2 = ax1.twinx()
ax2.set_ylabel("Execution Time (s)", color='tab:red')
ax2.plot(execution_methods, execution_times, marker='s', linestyle='--', color='tab:red', label="Execution Time")
ax2.tick_params(axis='y', labelcolor='tab:red')

# Add legends
ax1.legend(loc='upper right', handles=ax1.lines + ax2.lines, labels=["$\Delta V$", "Execution Time"])

# Improve grid and ticks
ax1.grid(which='major', color='#DDDDDD', linewidth=0.8)
ax1.grid(which='minor', color='#EEEEEE', linestyle=':', linewidth=0.5)

ax1.minorticks_on()
ax2.minorticks_on()
ax1.tick_params(axis='both', which='major', labelsize=12)
ax2.tick_params(axis='both', which='major', labelsize=12)

ax1.tick_params(axis='x', which='minor', length=0)

ax1.set_ylim(9.3,10.4)
#ax2.set_ylim(0,500)

# Show title
plt.title("$\Delta V$ and Execution Time")
plt.tight_layout()
plt.show()

# Get methods without "REAL" for execution time plotting
execution_methods = [m for m in methods if m in execution_time_dict]
execution_times = [execution_time_dict[name] for name in execution_methods]

# Get all methods for deltaV plotting
deltaV_values_method = [deltaVs[name] for name in methods]

# Convert to numpy arrays for plotting
deltaV_values_method = np.array(deltaV_values_method, dtype=float)
execution_times = np.array(execution_times, dtype=float)

fig, ax1 = plt.subplots(figsize=(10, 6))

# Plot delta-V with markers
ax1.set_xlabel("Optimisation Method", labelpad=10, fontsize=12)
ax1.set_ylabel("$\Delta V$ (km/s)", color='tab:blue', fontsize=12)
bars1 = ax1.bar(np.arange(len(methods)) - 0.2, deltaV_values_method, width=0.4, 
               color='tab:blue', alpha=0.7, label="$\Delta V$")
ax1.tick_params(axis='y', labelcolor='tab:blue')
ax1.set_xticks(range(len(methods)))
ax1.set_xticklabels(methods, ha='center', fontsize=12)
ax1.set_ylim(9.3, 10.4)

# Add data labels to bars
for i, bar in enumerate(bars1):
    height = bar.get_height()
    ax1.text(bar.get_x() + bar.get_width()/2., height + 0.02,
             f'{deltaV_values_method[i]:.2f}',
             ha='center', va='bottom', color='tab:blue', fontsize=10)

# Secondary axis for execution time (logarithmic scale)
ax2 = ax1.twinx()
ax2.set_ylabel("Execution Time (s, log scale)", color='tab:red', fontsize=12)

# Find the indices of execution methods in the original methods list
exec_indices = [methods.index(m) for m in execution_methods]
bars2 = ax2.bar(np.array(exec_indices) + 0.2, execution_times, width=0.4, 
               color='tab:red', alpha=0.7, label="Execution Time")
ax2.set_yscale('log')  # Use logarithmic scale
ax2.tick_params(axis='y', labelcolor='tab:red')

# Add data labels to execution time bars
for i, bar in enumerate(bars2):
    height = bar.get_height()
    ax2.text(bar.get_x() + bar.get_width()/2., height * 1.1,
             f'{execution_times[i]:.2f}s',
             ha='center', va='bottom', color='tab:red', fontsize=10)

# With this updated code:
handles = [bars1[0], bars2[0]]  # Get the first bar from each set
labels = ['$\Delta V$ (km/s)', 'Execution Time (s)']
ax1.legend(handles, labels, loc='upper right', fontsize=12)

# Improve grid and ticks
#ax1.grid(which='major', axis='y', color='#DDDDDD', linewidth=0.8, alpha=0.5)
ax1.tick_params(axis='both', which='major', labelsize=12)
ax2.tick_params(axis='both', which='major', labelsize=12)

ax1.set_ylim(9.3,10.4)
ax2.set_ylim(5e-1,10**3)

# Show title
plt.title(r"\textbf{Comparison of $\Delta V$ and Execution Time by Optimisation Method}", fontsize=14, pad=10)
plt.tight_layout()

output_figures.save_file("voyager_I_time_execution_comparison.png", dpi=500, bbox_inches="tight")
plt.show()

Figure saved to: Voyager 1 - Figures/voyager_I_time_execution_comparison.png

#math.factorial(20)

Variation in Heliocentric Velocity along Voyager I Trajectory

Voyager I NASA Mission Data

Using, https://plotdigitizer.com/app on the real mission NASA data presented in textbook Hahn-A-Basic-Calculus-of-Planetary-Orbits-and-Interplanetary-Flight-2020-pdf.

file_path = output_pickle.get_path('V1.csv')
data1 = pd.read_csv(file_path)

data1['x_km'] = data1['x'] * 149597870.691 

V1_NASA_distance_from_Sun_km = data1['x_km']
V1_NASA_distance_from_Sun_AU = data1['x']  # in AU where 1 AU = 149597870.691 km
V1_NASA_velocity = data1[' y']

#data1

r_esc = np.linspace(1e-99,6e9,1000)
v_esc = np.sqrt(2 * mu_sun/r_esc)

Comparison with Algorithms

# Known parameters K = [e,a,i,Omega,w,M]
mu = 1.327e11  # Sun's gravitational parameter (km³/s²)

# Compute true anomaly (theta) for 0<e<1 (elliptical orbit)
def elliptic_theta(M, e):
    def find_E(E):
        return E - e * np.sin(E) - M

    E = fsolve(find_E, 0.01)[0] # Eccentric Anomaly
    theta = 2 * np.arctan(np.tan(E / 2) * np.sqrt((1 + e) / (1 - e)))
    return theta


# Compute true anomaly (theta)  for e=1 (parabolic orbit)
def parabolic_theta(M):
    var = 3 * M + np.sqrt(9 * M**2 + 1)
    B = var ** (1 / 3) - var ** (-1 / 3) # Barker Parameter
    theta = 2 * np.arctan(B)
    return theta


# Compute true anomaly (theta)  for e>1 (hyperbolic orbit)
def hyperbolic_theta(M, e):
    def find_F(F):
        return e * np.sinh(F) - F - M

    F = fsolve(find_F, 0.01)[0] # Hyperbolic Eccentric Anomaly
    theta = 2 * np.arctan(np.tanh(F / 2) * np.sqrt((e + 1) / (e - 1)))
    return theta

def forwardTransformation(e, a, i, Omega, w, M, mu, p=None):
    # Compute semi-latus rectum p if not provided
    if p is None:
        if e < 1:
            p = a * (1 - e**2)
        elif e == 1:
            p = 2 * a  # For parabolic orbits, p = 2q where q is periapsis distance
        else:  # e > 1
            p = a * (e**2 - 1)  # For hyperbolic orbits
    
    # Verify that p is positive
    if p <= 0:
        raise ValueError(f"Semi-latus rectum p must be positive, got {p}")
    
    # Compute specific angular momentum h
    h = np.sqrt(mu * p)
    
    # Compute true anomaly from mean anomaly based on orbit type
    if e < 1:
        theta = elliptic_theta(M, e)
    elif e == 1:
        theta = parabolic_theta(M)
    else:
        theta = hyperbolic_theta(M, e)
    
    # Compute radius
    r = p / (1 + e * np.cos(theta))
    
    # Compute position and velocity in perifocal coordinates
    r_perifocal = np.array([r * np.cos(theta), r * np.sin(theta), 0])
    v_perifocal = (mu / h) * np.array([-np.sin(theta), e + np.cos(theta), 0])
    
    # Compute transformation matrix Q
    cos_Omega, sin_Omega = np.cos(Omega), np.sin(Omega)
    cos_w, sin_w = np.cos(w), np.sin(w)
    cos_i, sin_i = np.cos(i), np.sin(i)
    
    Q = np.array([
        [cos_Omega * cos_w - sin_Omega * sin_w * cos_i,
         -cos_Omega * sin_w - sin_Omega * cos_i * cos_w,
         sin_Omega * sin_i],
        [sin_Omega * cos_w + cos_Omega * cos_i * sin_w,
         -sin_Omega * sin_w + cos_Omega * cos_i * cos_w,
         -cos_Omega * sin_i],
        [sin_i * sin_w, sin_i * cos_w, cos_i]
    ])
    
    # Transform to reference frame
    r_reference = np.dot(Q, r_perifocal)
    v_reference = np.dot(Q, v_perifocal)
    
    return r_reference, v_reference

def backwardTransformation(r, v, mu):
    # Ensure inputs are numpy arrays
    r = np.array(r)
    v = np.array(v)
    
    # Compute magnitude of position vector
    r_mag = np.linalg.norm(r)
    
    # 1. Compute angular momentum vector h
    h_vec = np.cross(r, v)
    h_mag = np.linalg.norm(h_vec)
    
    # 2. Compute eccentricity vector
    e_vec = np.cross(v, h_vec) / mu - r / r_mag
    e_mag = np.linalg.norm(e_vec)
    
    # 3. Compute semi-latus rectum (parameter)
    p = h_mag**2 / mu
    
    # 4. Compute semi-major axis
    a = p / (1 - e_mag**2)
    
    # 5. Compute inclination using arccos(h_z/h)
    i = np.arccos(h_vec[2] / h_mag)
    
    # 6. Compute longitude of ascending node using atan2(h_x, -h_y)
    Omega = np.arctan2(h_vec[0], -h_vec[1]) % (2 * np.pi)
    
    # 7. Compute the argument of periapsis
    w = np.arctan2(
        e_vec[2] / np.sin(i),
        e_vec[1] * np.sin(Omega) + e_vec[0] * np.cos(Omega)
    ) % (2 * np.pi) # Use atan2 for quadrant disambiguation
    
    # 8. Compute true anomaly
    # Compute cos(theta) = r⋅e/(r*e)
    cos_theta = np.dot(r, e_vec) / (r_mag * e_mag)
    
    # Calculate theta and handle quadrant ambiguity
    theta = np.arccos(cos_theta)
    
    # Apply correction: 2π - θ if r⋅v < 0
    if np.dot(r, v) < 0:
        theta = 2 * np.pi - theta
    
    return p, e_mag, i, Omega, w, theta

import numpy as np
from scipy.optimize import fsolve
from poliastro.core.elements import rv2coe, coe2rv

def position_velocity_at_time(r1, v1, mu, t):
    
    # Function to convert true anomaly to mean anomaly (elliptical case)
    def true_to_mean_elliptic(theta, e):
        # First convert true anomaly to eccentric anomaly
        E = 2 * np.arctan(np.sqrt((1 - e) / (1 + e)) * np.tan(theta / 2))
        # Then convert eccentric anomaly to mean anomaly
        M = E - e * np.sin(E)
        return M
    
    # Function to convert true anomaly to mean anomaly (parabolic case)
    def true_to_mean_parabolic(theta):
        # Convert true anomaly to parabolic anomaly (Barker's parameter)
        B = np.tan(theta / 2)
        # Convert to mean anomaly equivalent
        M = (1/3) * B**3 + B
        return M
    
    # Function to convert true anomaly to mean anomaly (hyperbolic case)
    def true_to_mean_hyperbolic(theta, e):
        # Convert true anomaly to hyperbolic eccentric anomaly
        F = 2 * np.arctanh(np.sqrt((e - 1) / (e + 1)) * np.tan(theta / 2))
        # Convert hyperbolic eccentric anomaly to mean anomaly
        M = e * np.sinh(F) - F
        return M
    
    # Convert state vectors to orbital elements
    # Using standard notation: [p, e, i, Omega, w, theta]
    p, e, i, Omega, w, theta = backwardTransformation(r1, v1, mu) #rv2coe(mu, r1, v1)
    
    # Calculate semi-major axis
    if e < 1.0:  # Elliptical orbit
        a = p / (1 - e**2)
        
        # Convert true anomaly to mean anomaly
        M0 = true_to_mean_elliptic(theta, e)
        
        # Calculate mean motion
        n = np.sqrt(mu / a**3)
        
        # Propagate mean anomaly
        M = M0 + n * t
        
        # Convert back to true anomaly using provided function
        theta_new = elliptic_theta(M, e)
        
    elif abs(e - 1.0) < 1e-10:  # Parabolic case
        # For parabolic orbits, use Barker's equation
        # Parameter for parabolic orbit mean motion
        D = np.sqrt(mu / (2 * p**3))
        
        # Calculate the initial mean anomaly equivalent
        M0 = true_to_mean_parabolic(theta)
        
        # Propagate to new mean anomaly equivalent
        M = M0 + D * t
        
        # Convert back to true anomaly using provided function
        theta_new = parabolic_theta(M)
        
    else:  # Hyperbolic orbit
        a = p / (e**2 - 1)  # Negative for hyperbolic orbits
        
        # Convert true anomaly to mean anomaly for hyperbolic case
        M0 = true_to_mean_hyperbolic(theta, e)
        
        # Calculate mean motion
        n = np.sqrt(mu / abs(a)**3)
        
        # Propagate mean anomaly
        M = M0 + n * t
        
        # Convert back to true anomaly using provided function
        theta_new = hyperbolic_theta(M, e)
    
    # Convert orbital elements back to position and velocity vectors
    #r, v = coe2rv(mu, p, e, i, Omega, w, theta_new)
    r, v = forwardTransformation(e, a, i, Omega, w, M, mu, p)
    
    return r, v

def get_body_from_name(body_name):
    """Get poliastro body object from name"""
    bodies = {
        'sun': Sun,
        'mercury': Mercury,
        'venus': Venus,
        'earth': Earth,
        'mars': Mars,
        'jupiter': Jupiter,
        'saturn': Saturn,
        'uranus': Uranus,
        'neptune': Neptune
    }
    body_name = body_name.lower()
    
    return bodies[body_name]

def plot_transfer_velocity(date1, date2, planet1, planet2, current_tof, show_plot, num_points, prev_final_pos=None):
    
    # Convert dates to Time objects
    t0 = Time(Time(date1, scale="tt").tdb)
    tf = Time(Time(date2, scale="tt").tdb)
    
    # Get planet bodies
    body1 = get_body_from_name(planet1)
    body2 = get_body_from_name(planet2)
    
    # Calculate positions of planets at departure and arrival
    body1_ephem  = Ephem.from_body(body1, time_range(t0, end=tf, periods=500))#, plane=Planes.EARTH_ECLIPTIC)
    body2_ephem = Ephem.from_body(body2, time_range(t0, end=tf, periods=500))#, plane=Planes.EARTH_ECLIPTIC)
    
    r1 = body1_ephem.rv(t0)[0].to(u.km).value
    r2 = body2_ephem.rv(tf)[0].to(u.km).value

    v1_p = body1_ephem.rv(t0)[1].to(u.km / u.s).value
    v2_p = body2_ephem.rv(tf)[1].to(u.km / u.s).value
    v1_p_norm = np.linalg.norm(body1_ephem.rv(t0)[1].to(u.km / u.s).value)
    v2_p_norm = np.linalg.norm(body2_ephem.rv(tf)[1].to(u.km / u.s).value)
    
    # Time of flight in seconds
    tof = (tf - t0).to(u.s).value
    
    # Calculate initial and final velocities using Lambert's problem
    v1, v2 = lambert(r1, r2, tof)

    v1_p_sc = np.linalg.norm(v1 + v1_p)
    v2_p_sc = np.linalg.norm(v2 + v2_p)
    
    # Time points for the trajectory
    times = np.linspace(0, tof, num_points)
    
    # Calculate velocity magnitude at each time point
    velocities = []
    positions = []
    for t in times:
        r, v = position_velocity_at_time(r1, v1, mu_sun, t)
        positions.append(r)
        velocities.append(np.linalg.norm(v))

    times = np.linspace(0 + current_tof, tof + current_tof, num_points)
    #if planet1 == "jupiter":
    #    positions = np.array(positions) + 15028472.596829772
        
    # Convert to days
    days = times / (24 * 3600)
    positions_norm_km = np.array([np.linalg.norm(pos) for pos in positions])
    positions_norm_AU = np.array([np.linalg.norm(pos) for pos in positions]) / 149597870.691
    
    return days, positions, velocities, v1, v2, tof, r1, r2, v1_p_sc, v2_p_sc

Graph

colours = {
    "Real Mission": "crimson",     
    "Brute Force Algorithm": "cyan", 
    "FMIN": "magenta",                 
    "GA": "gold",                      
    "GA --> FMIN": "lime",         
}

# Distinct line styles to prevent visual merging
line_styles = {
    "Real Mission": "-",              
    "Brute Force Algorithm": "-",    
    "FMIN": "--",                     
    "GA": '--',#(0, (5, 1)),               
    "GA --> FMIN": ":",              
}

# Define different line widths for clarity
line_widths = {
    "Real Mission": 2,
    "Brute Force Algorithm": 4,
    "FMIN": 3,
    "GA": 2,
    "GA --> FMIN": 2
}

# Setup LaTeX rendering
plt.rc('text', usetex=True)
plt.rc('font', family='serif')

# Create figure with primary axis
fig, ax1 = plt.subplots(figsize=(9, 7))

def plot_voyager_trajectory_vs_distance(date_departure, date_flyby_jupiter, date_flyby_saturn, method, plotlabel):
    flag = False  # Don't show individual plots
    print("Method:", method)

    planet1 = "earth"
    planet2 = "jupiter"
    
    days1, positions1, velocities1, v1, v2, tof_to_jupiter, r1, r2, v1_p_sc, v2_p_sc = plot_transfer_velocity(
        date1 = date_departure, 
        date2 = date_flyby_jupiter,
        planet1 = "earth", 
        planet2 = "jupiter",
        current_tof = 0,
        show_plot = flag, 
        num_points = 100
    )
    
    print(f"[{planet1} to {planet2}] Initial velocity: {np.linalg.norm(v1):.2f} km/s")
    print(f"[{planet1} to {planet2}] Final velocity: {np.linalg.norm(v2):.2f} km/s")
    print(f"[{planet1} to {planet2}] Initial Sun Distance: {np.linalg.norm(r1):.2e} km")
    print(f"[{planet1} to {planet2}] Final Sun Distance: {np.linalg.norm(r2):.2e} km")

    planet1 = "jupiter"
    planet2 = "saturn"

    days2, positions2, velocities2, v1, v2, tof_to_saturn, r1, r2, v1_p_sc_jupiter, v2_p_sc_jupiter = plot_transfer_velocity(
        date1 = date_flyby_jupiter, 
        date2 = date_flyby_saturn,
        planet1 = "jupiter", 
        planet2 = "saturn",
        current_tof = tof_to_jupiter,
        show_plot = flag, 
        num_points = 100
    )
    
    print(f"\n[{planet1} to {planet2}] Initial velocity: {np.linalg.norm(v1):.2f} km/s")
    print(f"[{planet1} to {planet2}] Final velocity: {np.linalg.norm(v2):.2f} km/s")
    print(f"[{planet1} to {planet2}] Initial Sun Distance: {np.linalg.norm(r1):.2e} km")
    print(f"[{planet1} to {planet2}] Final Sun Distance: {np.linalg.norm(r2):.2e} km")
    print("--------------------------------------------------------")
    
    # Calculate distances from the Sun in km
    distances1_km = np.array([np.linalg.norm(pos) for pos in positions1])
    distances2_km = np.array([np.linalg.norm(pos) for pos in positions2])
    
    # Concatenate distances and velocities - in km
    distances_final = distances1_km.tolist() + [np.linalg.norm(positions1[-1])] + distances2_km.tolist()
    velocities_final = velocities1 + [v1_p_sc_jupiter] + velocities2
    
    # Plot velocity vs distance
    ax1.plot(distances_final, velocities_final,
             color=colours[method], 
             label=plotlabel,
             linestyle=line_styles[method],
             linewidth=line_widths[method])
    
    # Return for potential further use
    return distances_final, velocities_final

# Execute plotting for each mission
# plot_voyager_trajectory_vs_distance(
#     date_departure,
#     date_flyby_jupiter,
#     date_flyby_saturn,
#     method="Real Mission",
#     plotlabel="Real Mission"
# )

ax1.plot(V1_NASA_distance_from_Sun_km, V1_NASA_velocity,
         color=colours["Real Mission"], 
         label="Real Mission",
         linestyle=line_styles["Real Mission"],
         linewidth=line_widths["Real Mission"])

plot_voyager_trajectory_vs_distance(
    date_departure_BRUTE_MIN,
    date_flyby_jupiter_BRUTE_MIN,
    date_flyby_saturn_BRUTE_MIN,
    method="Brute Force Algorithm",
    plotlabel="Brute Force Search"
)

plot_voyager_trajectory_vs_distance(
    date_departure_FMIN,
    date_flyby_jupiter_FMIN,
    date_flyby_saturn_FMIN,
    method="FMIN",
    plotlabel="Fmin"
)

plot_voyager_trajectory_vs_distance(
    date_departure_GA,
    date_flyby_jupiter_GA,
    date_flyby_saturn_GA,
    method="GA",
    plotlabel="Genetic Algorithm"
)

plot_voyager_trajectory_vs_distance(
    date_departure_GA_FMIN_COMBINED,
    date_flyby_jupiter_GA_FMIN_COMBINED,
    date_flyby_saturn_GA_FMIN_COMBINED,
    method="GA --> FMIN",
    plotlabel="GA + Fmin"
)

# Add Solar System escape velocity if available
try:
    ax1.plot(r_esc, v_esc, ':', label="Solar System $V_{esc}$", color='gray', linewidth=1.5)
except NameError:
    pass

# Labels for planets (adjust positions as needed based on your data)
earth_distance = 1.5e8  # Earth's distance from Sun in km
jupiter_distance = 7.78e8  # Jupiter's distance from Sun in km
saturn_distance = 1.43e9  # Saturn's distance from Sun in km

# ax1.annotate('Earth', xy=(earth_distance, 30), xytext=(earth_distance, 41), fontsize=12, color='black')
# ax1.annotate('Jupiter', xy=(jupiter_distance, 25), xytext=(jupiter_distance, 37.5), fontsize=12, color='black')
# ax1.annotate('Saturn', xy=(saturn_distance, 20), xytext=(saturn_distance, 34), fontsize=12, color='black')

# Primary x-axis (Distance in km)
ax1.set_xlabel("Distance from the Sun (km)")
ax1.set_ylabel("Heliocentric Velocity (Sun-centred) (km/s)")
ax1.set_ylim(0, 45)

# Adjust the x-axis limits to show relevant range
ax1.set_xlim(1e8, 1.6e9)  # This covers Earth to slightly beyond Saturn

# Create secondary x-axis (Distance in AU)
ax2 = ax1.twiny()
ax2.set_xlabel("Distance from the Sun (AU)")

# Set tick positions based on AU
km_to_AU = 1 / 149597870.691  # Conversion factor
ax2.set_xlim(ax1.get_xlim()[0] * km_to_AU, ax1.get_xlim()[1] * km_to_AU)

# Improve legend
ax1.legend(loc="best", fontsize=12, frameon=True, framealpha=0.9, edgecolor='black')

# Improve grid and ticks
ax1.grid(which='major', color='#DDDDDD', linewidth=0.8)
ax1.grid(which='minor', color='#EEEEEE', linestyle=':', linewidth=0.5)
ax1.minorticks_on()

# Customize tick parameters
ax1.tick_params(axis='both', which='major', labelsize=12)
ax1.tick_params(axis='both', which='minor', labelsize=10)
ax2.tick_params(axis='both', which='major', labelsize=12)

plt.title(r'\textbf{Voyager I: Heliocentric Velocity vs Distance from the Sun}', fontsize=14, pad=10)
plt.tight_layout()

output_figures.save_file('Voyager_I_Heliocentric_Velocity_vs_Distance.png', dpi=500, bbox_inches="tight")

plt.show()

Method: Brute Force Algorithm
[earth to jupiter] Initial velocity: 38.79 km/s
[earth to jupiter] Final velocity: 9.25 km/s
[earth to jupiter] Initial Sun Distance: 1.52e+08 km
[earth to jupiter] Final Sun Distance: 7.98e+08 km

[jupiter to saturn] Initial velocity: 19.03 km/s
[jupiter to saturn] Final velocity: 14.63 km/s
[jupiter to saturn] Initial Sun Distance: 7.98e+08 km
[jupiter to saturn] Final Sun Distance: 1.44e+09 km
--------------------------------------------------------
Method: FMIN
[earth to jupiter] Initial velocity: 38.78 km/s
[earth to jupiter] Final velocity: 9.17 km/s
[earth to jupiter] Initial Sun Distance: 1.51e+08 km
[earth to jupiter] Final Sun Distance: 7.98e+08 km

[jupiter to saturn] Initial velocity: 18.89 km/s
[jupiter to saturn] Final velocity: 14.45 km/s
[jupiter to saturn] Initial Sun Distance: 7.98e+08 km
[jupiter to saturn] Final Sun Distance: 1.44e+09 km
--------------------------------------------------------
Method: GA
[earth to jupiter] Initial velocity: 38.78 km/s
[earth to jupiter] Final velocity: 9.17 km/s
[earth to jupiter] Initial Sun Distance: 1.51e+08 km
[earth to jupiter] Final Sun Distance: 7.98e+08 km

[jupiter to saturn] Initial velocity: 18.89 km/s
[jupiter to saturn] Final velocity: 14.45 km/s
[jupiter to saturn] Initial Sun Distance: 7.98e+08 km
[jupiter to saturn] Final Sun Distance: 1.44e+09 km
--------------------------------------------------------
Method: GA --> FMIN
[earth to jupiter] Initial velocity: 38.78 km/s
[earth to jupiter] Final velocity: 9.17 km/s
[earth to jupiter] Initial Sun Distance: 1.51e+08 km
[earth to jupiter] Final Sun Distance: 7.98e+08 km

[jupiter to saturn] Initial velocity: 18.89 km/s
[jupiter to saturn] Final velocity: 14.45 km/s
[jupiter to saturn] Initial Sun Distance: 7.98e+08 km
[jupiter to saturn] Final Sun Distance: 1.44e+09 km
--------------------------------------------------------
Figure saved to: Voyager 1 - Figures/Voyager_I_Heliocentric_Velocity_vs_Distance.png

DeltaV to Fuel Mass

def compute_fuel_mass(M0, Vex, deltaV):
    Mf = M0*( 1 - np.exp(-deltaV/Vex) )
    return Mf 

def mission_fuel_mass(deltaV_injection, deltaV_jupiter):
    M0_1 = M0
    Mf_1 = compute_fuel_mass(M0_1, Vex, deltaV_injection)
    #print(Mf_1)
    M0_2 = M0_1 - Mf_1
    #print(M0_2)
    Mf_2 = compute_fuel_mass(M0_2, Vex, deltaV_jupiter)

    Mf_total = Mf_1 + Mf_2

    return Mf_total

deltaV_injection_values = {
    "REAL": data_dict_REAL['deltaV_injection (km/s)'],
    "Brute Force": data_dict_BRUTE_MIN['deltaV_injection (km/s)'],
    "Fmin": data_dict_FMIN['deltaV_injection (km/s)'],
    "Genetic Algorithm": data_dict_GA['deltaV_injection (km/s)'],
}

deltaV_jupiter_values = {
    "REAL": data_dict_REAL['deltaV_jupiter (km/s)'],
    "Brute Force": data_dict_BRUTE_MIN['deltaV_jupiter (km/s)'],
    "Fmin": data_dict_FMIN['deltaV_jupiter (km/s)'],
    "Genetic Algorithm": data_dict_GA['deltaV_jupiter (km/s)'],
}

# Compute and print fuel masses for each method
fuel_masses = {}
for method in deltaV_injection_values:
    fuel_masses[method] = mission_fuel_mass(deltaV_injection_values[method], deltaV_jupiter_values[method])
    print(f"{method}: {fuel_masses[method]:.2f} kg of fuel used")

REAL: 1139.00 kg of fuel used
Brute Force: 1071.75 kg of fuel used
Fmin: 1069.96 kg of fuel used
Genetic Algorithm: 1069.96 kg of fuel used

# Given deltaVs
deltaVss = {
    "REAL": deltaV_REAL,
    "Brute Force": deltaV_BRUTE_MIN,
    "Fmin": deltaV_FMIN,
    "Genetic Algorithm": deltaV_GA
}

fuel_masses = {}
for method in deltaVss:
    fuel_masses[method] = compute_fuel_mass(M0, Vex, deltaVss[method])

# Print results
for method, fuel_mass in fuel_masses.items():
    print(f"{method}: {fuel_mass:.2f} kg of fuel used")

REAL: 1139.00 kg of fuel used
Brute Force: 1071.75 kg of fuel used
Fmin: 1069.96 kg of fuel used
Genetic Algorithm: 1069.96 kg of fuel used

deltaVss # in km/s

{'REAL': 10.31241583834745,
 'Brute Force': 9.411191461145675,
 'Fmin': 9.388030073904538,
 'Genetic Algorithm': 9.388030185988406}

Voyager I & II NASA Mission Data

file_path = output_pickle.get_path('V1.csv')
data1 = pd.read_csv(file_path)

data1['x_km'] = data1['x'] * 149597870.691 

V1_NASA_distance_from_Sun_km = data1['x_km']
V1_NASA_distance_from_Sun_AU = data1['x']  # in AU where 1 AU = 149597870.691 km
V1_NASA_velocity = data1[' y']

data1

	x	y	x_km
0	0.895445	39.838746	1.339567e+08
1	1.073111	37.004831	1.605351e+08
2	1.243803	34.356325	1.860703e+08
3	1.412753	31.459340	2.113448e+08
4	1.726100	28.346305	2.582209e+08
5	2.014462	25.878735	3.013592e+08
6	2.338850	23.657631	3.498870e+08
7	2.895525	20.453570	4.331644e+08
8	3.494765	17.764582	5.228094e+08
9	4.139764	15.619520	6.192999e+08
10	5.180000	13.097155	7.749170e+08
11	5.180000	36.097021	7.749170e+08
12	5.200000	23.637279	7.779089e+08
13	6.138537	22.456611	9.183121e+08
14	7.602425	21.186482	1.137307e+09
15	8.709383	20.475264	1.302905e+09
16	9.430000	20.301038	1.410708e+09
17	9.430000	23.056674	1.410708e+09
18	9.450000	21.542986	1.413700e+09
19	10.375342	21.183351	1.552129e+09
20	11.289237	20.863079	1.688846e+09
21	13.146080	20.289184	1.966626e+09
22	15.106646	19.849034	2.259922e+09
23	17.439395	19.486268	2.608896e+09
24	19.618448	19.128198	2.934878e+09
25	22.337760	18.834094	3.341681e+09
26	25.753644	18.572195	3.852690e+09
27	28.773230	18.397075	4.304414e+09
28	32.154829	18.202272	4.810294e+09
29	35.625769	18.050188	5.329539e+09
30	37.961278	17.993827	5.678926e+09
31	39.998402	17.890276	5.983676e+09

file_path = output_pickle.get_path('V2.csv')
data2 = pd.read_csv(file_path)

data2['x_km'] = data2['x'] * 149597870.691 

V2_NASA_distance_from_Sun_km = data2['x_km']
V2_NASA_distance_from_Sun_AU = data2['x']  # in AU where 1 AU = 149597870.691 km
V2_NASA_velocity = data2[' y']

data2

	x	y	x_km
0	0.982667	39.034907	1.470049e+08
1	1.106842	36.829962	1.655812e+08
2	1.281253	34.222948	1.916728e+08
3	1.459839	31.694050	2.183888e+08
4	1.671518	28.857742	2.500555e+08
5	1.922849	26.352738	2.876541e+08
6	2.232168	23.640037	3.339277e+08
7	2.719179	20.187530	4.067834e+08
8	3.326190	17.032554	4.975910e+08
9	3.962270	14.320542	5.927472e+08
10	4.654699	11.697905	6.963330e+08
11	5.262158	9.897326	7.872076e+08
12	5.322829	26.395472	7.962839e+08
13	5.331177	23.528607	7.975327e+08
14	5.397811	20.146174	8.075010e+08
15	6.286713	18.403723	9.404789e+08
16	7.631321	16.802800	1.141629e+09
17	8.866661	15.844498	1.326434e+09
18	9.550741	15.517329	1.428771e+09
19	9.604406	32.337590	1.436799e+09
20	9.626022	20.284486	1.440032e+09
21	11.326906	19.549734	1.694481e+09
22	13.759424	18.874488	2.058381e+09
23	16.755574	18.156738	2.506598e+09
24	19.146353	17.880115	2.864254e+09
25	19.175272	21.892761	2.868580e+09
26	19.164539	19.718833	2.866974e+09
27	21.597206	19.384541	3.230896e+09
28	24.233055	19.119635	3.625213e+09
29	27.971572	18.977418	4.184488e+09
30	30.267841	18.857027	4.528005e+09
31	30.253381	28.403059	4.525841e+09
32	30.336413	16.708831	4.538263e+09
33	33.753834	16.590737	5.049502e+09
34	36.886085	16.423477	5.518080e+09
35	40.072299	16.341684	5.994731e+09

r_esc = np.linspace(1e-99,6e9,1000)
v_esc = np.sqrt(2 * mu_sun/r_esc)

plt.rc('text', usetex=True)
plt.rc('font', family='serif')

fig, ax1 = plt.subplots(figsize=(9, 7))

# Plot Voyager I and II trajectories
ax1.plot(V1_NASA_distance_from_Sun_km, V1_NASA_velocity, label="Voyager I", color='cyan', linewidth=1.5)
ax1.plot(V2_NASA_distance_from_Sun_km, V2_NASA_velocity, label="Voyager II", color='red', linewidth=1.5)

# Plot Solar System escape velocity
ax1.plot(r_esc, v_esc, '--', label="Solar System $V_{esc}$", color='gray', linewidth=1)

# Labels for planets
ax1.annotate('Earth', xy=(1.5e8, 40), xytext=(0, 41), fontsize=12, color='black')
ax1.annotate('Jupiter', xy=(7.78e8, 40), xytext=(5.5e8, 37.5), fontsize=12, color='black')
ax1.annotate('Saturn', xy=(1.43e9, 40), xytext=(1.225e9, 34), fontsize=12, color='black')
ax1.annotate('Uranus', xy=(2.87e9, 40), xytext=(2.65e9, 23.5), fontsize=12, color='black')
ax1.annotate('Neptune', xy=(4.50e9, 40), xytext=(4.275e9, 30), fontsize=12, color='black')

# Primary x-axis (Distance in km)
ax1.set_xlabel("Distance from the Sun (km)", fontsize=12)
ax1.set_ylabel("Heliocentric Velocity (Sun-centred) (km/s)", fontsize=12)
ax1.set_ylim(0, 45)

# Create secondary x-axis (Distance in AU)
ax2 = ax1.twiny()
ax2.set_xlabel("Distance from the Sun (AU)", fontsize=12)

# Set tick positions based on AU
km_to_AU = 1 / 149597870.691  # Conversion factor
ax2.set_xlim(ax1.get_xlim()[0] * km_to_AU, ax1.get_xlim()[1] * km_to_AU)

# Improve legend
ax1.legend(loc="best", fontsize=12, frameon=True, framealpha=0.9, edgecolor='black')

# Improve grid and ticks
ax1.grid(which='major', color='#DDDDDD', linewidth=0.8)
ax1.grid(which='minor', color='#EEEEEE', linestyle=':', linewidth=0.5)
ax1.minorticks_on()

# Customize tick parameters
ax1.tick_params(axis='both', which='major', labelsize=12)
ax1.tick_params(axis='both', which='minor', labelsize=10)
ax2.tick_params(axis='both', which='major', labelsize=12)

plt.title(r"\textbf{Heliocentric Velocity vs Distance from the Sun}", fontsize=14, pad=10)

plt.tight_layout()
output_figures.save_file('Voyager_1_and_2_VvsDistance.png', dpi=500, bbox_inches="tight")
plt.show()

Figure saved to: Voyager 1 - Figures/Voyager_1_and_2_VvsDistance.png